https://mathpuzzlewiki.com/api.php?action=feedcontributions&user=Zerrakhi&feedformat=atomMath Puzzle Wiki - User contributions [en]2021-08-04T08:15:17ZUser contributionsMediaWiki 1.29.0https://mathpuzzlewiki.com/index.php?title=Talk:Eight_options_with_six_sides&diff=750Talk:Eight options with six sides2010-12-24T02:40:59Z<p>Zerrakhi: /* This seems too easy */</p>
<hr />
<div>== This seems too easy ==<br />
<br />
Determining the <i>least</i> number of rolls he <i>will</i> have to make is so trivial I wouldn't even call it a puzzle. Determining the <i>greatest</i> number of rolls he <i>may</i> have to make is a slightly less trivial, but still very simple. One can optimise either of these values. Perhaps one for [[:Category:Easy puzzles]]?<br />
<br />
As far as I can tell, I'm not missing anything. [[User:Zerrakhi|Zerrakhi]] 03:43, 22 October 2010 (EDT)<br />
<br />
: Hmmm. I'm not seeing this as easy. Perhaps I am missing something. That or I worded the question in an unclear way. Could you tell me what you think the answer is? That would help me understand what is going on here. [[User:Oscarlevin|Oscarlevin]] 15:03, 18 November 2010 (EST)<br />
<br />
:: '''To optimise the LEAST number of rolls he WILL have to make:'''<br />
<br />
:: Let N be the least integer such that 6^N >= D where D is the number of dishes.<br />
:: Of the 6^N possible permutations of rolled numbers, allocate a certain number to each dish and leave the rest blank.<br />
:: The number of permutations allocated to each dish is the greatest integer A such that A <= (6^N)/D.<br />
:: Roll N dice. If the permutation that's rolled is allocated a dish, select that dish. Otherwise roll again.<br />
:: With this method, an infinite number of rolls MAY be needed, but the least number of rolls that WILL be needed is simply N.<br />
:: If D = 8 then N = 2 and A = 4.<br />
<br />
:: '''To optimise the MOST number of rolls he MAY have to make:'''<br />
<br />
:: Let N be the least integer such that 6^N is a multiple of D.<br />
:: Allocate permutations arbitrarily to dishes, as above except that now every permutation is allocated a dish (there are no blank spaces).<br />
:: With this method, exactly N rolls are needed, no more no less.<br />
:: If D = 8 then N = 3 and A = 27 (because 6 ^ 3 = 8 * 27).<br />
:: The least easy bit is characterising the values of D such that N exists, but that's not part of the main puzzle.<br />
<br />
:: [[User:Zerrakhi|Zerrakhi]] 21:40, 23 December 2010 (EST)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Around_the_world&diff=594Around the world2010-10-25T07:25:02Z<p>Zerrakhi: </p>
<hr />
<div>A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant speed (relative to the ground) and rate of fuel consumption, and that all planes return safely to their island base?<br />
<br />
{{Solution | [Update: I (User:Zerrakhi) have since learned that the following solution is wrong. There is a better and simpler solution that uses only three aircraft and relies on a step that I missed. But I don't have the heart to delete all my hard work.]<br />
<br />
First, let's agree on conventions. The main journey will be westward, and all units will be in degrees. So 360 degrees is a full circle around the earth, 1 unit of fuel is enough fuel to travel one degree, and 1 unit of time is the time taken to travel one degree.<br />
<br />
One unassisted aeroplane can travel 180 degrees non-returning, or 90 degrees returning. Let's see how far we can get with two aeroplanes: a main flight and an assisting flight.<br />
<br />
:The assisting flight requires fuel for the journey out, fuel for the transfer, and fuel for the return. The best place to transfer fuel to the main flight would be when the amount of fuel available (not needed for return) is exactly the amount of empty space in the main flight's fuel tanks. That way no fuel is wasted.<br />
<br />
:Space in tank <nowiki>=</nowiki> D (distance from base, since fuel used equals distance travelled)<br />
:Transferrable fuel <nowiki>=</nowiki> 180 - 2D (starting amount minus what's used for flight out and return)<br />
<br />
:Equating these:<br />
:D <nowiki>=</nowiki> 180 - 2D<br />
:D <nowiki>=</nowiki> 60<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 60, enabling it to reach 240 degrees non-returning (the 60 degrees already travelled plus a further 180 on a full tank), or 120 degrees returning (reserve 60 units of fuel for returning the 60 degrees already travelled, which leaves 120 units of fuel for travelling a further 60 units and back).<br />
<br />
With three aeroplanes, several scenarios are possible:<br />
*Two assisting flights both transfer fuel to the main flight at different locations on the latter's outward journey, the first at D <nowiki>=</nowiki> 60, the second later.<br />
*One flight transfers fuel to the primary assisting flight at D <nowiki>=</nowiki> 60, which in turn transfers fuel to the main flight.<br />
*Two assisting flights both transfer fuel to a main returning flight at D <nowiki>=</nowiki> 60, one on its flight out, one on its return.<br />
<br />
The first two scenarios are equivalent with respect to the optimal distance for the second transfer and the total distance that the main flight can travel. I will skip the first scenario and do the maths for the second.<br />
<br />
:Secondary assisting flight transfers fuel to primary assisting flight at D <nowiki>=</nowiki> 60.<br />
<br />
:We use the same logic as before to determine the optimal time to transfer fuel from the primary assisting flight to the main flight.<br />
<br />
:Space in main flight's tank <nowiki>=</nowiki> D<br />
:Transferrable fuel from primary assisting flight <nowiki>=</nowiki> 180 - 2(D - 60) - 60 (because tank full at D <nowiki>=</nowiki> 60 but need to travel that distance again on return)<br />
:This simplifies to 240 - 2D<br />
<br />
:D <nowiki>=</nowiki> 240 - 2D<br />
:D <nowiki>=</nowiki> 80<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 80, enabling it to reach 260 degrees non-returning or 130 degrees returning.<br />
<br />
As for the third scenario, the main flight is refueled at D <nowiki>=</nowiki> 60, enabling it to reach 150 degrees returning (90 degrees further than the 60 degrees already travelled), to be refueled again at D <nowiki>=</nowiki> 60 on its return. So this scenario achieves a longer return flight than the others at the cost of necessarily being a return flight.<br />
<br />
With four aeroplanes, we can combine the second and third scenarios above. One flight refuels the primary assisting flight at D <nowiki>=</nowiki> 60, which in turn refuels the main flight. The primary assisting flight is then refuelled by a fourth flight at D <nowiki>=</nowiki> 60 on its return.<br />
<br />
:Space in main flight's tank <nowiki>=</nowiki> D<br />
:Transferrable fuel from primary assisting flight <nowiki>=</nowiki> 180 - 2(D - 60) (this time we don't have to reserve the 60 units of fuel)<br />
:This simplifies to 300 - 2D<br />
<br />
:D <nowiki>=</nowiki> 300 - 2D<br />
:D <nowiki>=</nowiki> 100<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 100, enabling it to reach 280 degrees non-returning or 140 degrees returning.<br />
<br />
Now, 280 is only 80 short of 360 (a complete circumnavigation). Using two aeroplanes, one refueling the other, we can transport 80 units of transferable fuel 80 degrees from base. The second aeroplane, having been refueled at D <nowiki>=</nowiki> 60, travels a further 20 degrees out and must reserve enough fuel for 80 degrees return, leaving 80 units of fuel for the transfer.<br />
<br />
We can now piece together a solution. Three assisting flights travel west to enable the main flight to reach D <nowiki>=</nowiki> 280, and two more assisting flights travel east to enable the main flight to get from D <nowiki>=</nowiki> 280 to home.<br />
<br />
A more detailed itinery follows. The times (T) are easy to calculate given that the time is always equal to the location of the main flight, and that all flights travel at the same speed.<br />
<br />
*At T<nowiki>=</nowiki>0, main flight, western primary assisting flight, and 1st western secondary assisting flight begin westward journey.<br />
*At T<nowiki>=</nowiki>60, 1st western secondary assisting flight refills western primary assisting flight at D <nowiki>=</nowiki> 60.<br />
*At T<nowiki>=</nowiki>80, 2nd western secondary assisting flight begins westward journey.<br />
*At T<nowiki>=</nowiki>100, western primary assisting flight refills main flight at D <nowiki>=</nowiki> 100, giving it enough fuel to reach D <nowiki>=</nowiki> 280.<br />
*At T<nowiki>=</nowiki>120, 1st western secondary assisting flight returns to base.<br />
*At T<nowiki>=</nowiki>140, 2nd western secondary assisting flight partially refills returning western primary assisting flight at D <nowiki>=</nowiki> 60.<br />
*At T<nowiki>=</nowiki>200, western primary assisting flight and 2nd western secondary assisting flight return to base.<br />
*At T<nowiki>=</nowiki>200, eastern primary assisting flight and eastern secondary assisting flight begin eastward journey.<br />
*At T<nowiki>=</nowiki>260, eastern secondary assisting flight refills eastern primary assisting flight at D <nowiki>=</nowiki> 300 (60 from 360).<br />
*At T<nowiki>=</nowiki>280, eastern primary assisting flight partially refills main flight at D <nowiki>=</nowiki> 280 (80 from 360).<br />
*At T<nowiki>=</nowiki>320, eastern secondary assisting flight returns to base.<br />
*At T<nowiki>=</nowiki>360, main flight and eastern primary assisting flight return to base.<br />
<br />
This involves a total of six flights: one main, five returning. But by the time the eastern assisting flights have to depart, the western assisting flights have returned to the island. So the same aeroplanes can be re-used for the eastern assisting flights, reducing the number of aeroplanes needed by two.<br />
<br />
Therefore the puzzle can be solved with a total of four aeroplanes, including the main one. But certainly no fewer. Nothing is wasted in the above scenario, which confirms that it is the optimal solution. }}<br />
<br />
[[Category: Optimization puzzles]]<br />
[[Category: Geometry]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Help:Ways_to_contribute&diff=593Help:Ways to contribute2010-10-23T05:53:39Z<p>Zerrakhi: </p>
<hr />
<div>If you are interested in contributing to the Math Puzzle Wiki, but don't know what to do, here are some ideas:<br />
<br />
# Add a new puzzle. If you have a favorite brain teaser or math puzzle that doesn't appear to be on the site already, please add it. Even if you don't have all the details, at least get the page started, and hopefully someone else will clean it up.<br />
# Clean up a current puzzle. Many of the puzzle on this wiki were put up in a hurry. For example, I remembered it, and typed it up quickly so I wouldn't forget. Thus most all pages could be added to. The puzzle presentation itself might need to be improved, or perhaps there is a variation that should be mentioned. <br />
# Add a hint, answer or solution to a puzzle. Few puzzles on this site include hints/answers/solutions. If you have a good hint, or would like to add an answer or solution to any puzzle, please do. All I ask is that you use the [[Help:Hints,_answers_and_solutions | templates]] which will hide these sections by default, and require readers to log in to see. <br />
# Add a discussion of mathematical content. Many puzzles here illustrate mathematical concepts. Ideally this concepts should be at least mentioned on the puzzle page. Currently, they are not. Help fix that.<br />
# Improve cross-listing and navigation. Some puzzle pages include a list of similar puzzles. As new puzzles are added, these lists tend to fall out of date. Many pages have none at all. Additionally, each puzzle page should have a couple of categories to help direct readers to similar puzzles. These categories can describe either the type of puzzle (e.g. crossing puzzle, logic puzzle) or the mathematical basis for the puzzle (e.g., number theory, graph theory).<br />
# Improve the wiki itself. This is my first attempt at a wiki, and I welcome any suggestions. <br />
# Spread the word. The more math puzzle enthusiasts involved, the better.</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=592User talk:Oscarlevin2010-10-23T04:57:28Z<p>Zerrakhi: /* Link */</p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::: Good idea. I've started doing so.<br />
<br />
:::: Don't forget to add links to pages like that from the front page so people can find them... Also, what about adding the needs_hint, needs_answer, etc templates? Is that something people can usefully do? Since the templates exist, it seems a shame not to use them consistently. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
:::: (This section later trimmed of less important topics to enhance focus on what matters. See history for details.) [[User:Zerrakhi|Zerrakhi]] 00:23, 19 October 2010 (EDT)<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)<br />
<br />
::: The reason for trying to restrict solutions is that I want to make it difficult for students to use this site to cheat. I teach a variety of college courses where I use puzzles from this site as either extra credit assignments or as the beginnings of homework problems. If a student finds the site by searching for the puzzle, I would rather they feel intimidated by registering (because maybe they think I would see their email) and try to do the problem on their own. But I do want to have solutions somewhere to help remind me how difficult the puzzle is in case I forget how to solve it.<br />
<br />
::: I envision the site to sever as a collection of good math puzzles, with perhaps a discussion on why they are mathematically interesting. These puzzles are meant to puzzled over, not answered by a web search. Thoughts? (I probably should outline this "policy" on a help page as well.) [[User:Oscarlevin|Oscarlevin]] 22:03, 13 October 2010 (EDT)<br />
<br />
:::: Not a lot of relevant thoughts, but as you probably guessed, that was largely my reason for asking - to suggest things that would be useful to mention on the help pages. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
== Something's wrong... ==<br />
<br />
All of a sudden, I find that the hint/answer/solution boxes are '''open by default''', so that I can read their contents without wanting to! What could have caused this? (Added later: I'd say your edits to Common.js messed up the NavFrame definition somehow.) [[User:Zerrakhi|Zerrakhi]] 09:55, 18 October 2010 (EDT)<br />
<br />
: Exactly right. I undid the changes, so it should be working now. Thanks for the catch. I was trying to get the sidebar to have collapsible categories (haven't figured it out yet) and must have deleted the wrong thing. [[User:Oscarlevin|Oscarlevin]] 10:05, 18 October 2010 (EDT)<br />
<br />
:: Oddly, I've re-started my browser (to clear cookies, just in case) and still having the problem. <br />
<br />
:: (You can probably guess how I figured out the approximate cause. As puzzles go, it was easier than many on the site. Entered edit mode on solution template to see how it worked, searched common.js for mention of NavFrame, hunch confirmed.)<br />
<br />
::: Working now, though. Don't understand that. [[User:Zerrakhi|Zerrakhi]] 10:23, 18 October 2010 (EDT)<br />
<br />
:::: Must have been cached in your browser. There is a message on [[mediawiki:common.js]] about having to refresh while holding down SHIFT (for firefox anyway). [[User:Oscarlevin|Oscarlevin]] 10:25, 18 October 2010 (EDT)<br />
<br />
== Link ==<br />
<br />
Just want say that I've posted a [http://outerhoard.wordpress.com/2010/10/21/interesting-stuff-late-october-2010/ link] to the Math Puzzle Wiki on my blog.<br />
<br />
Also, you still haven't put the "[[MathPuzzleWiki:How_to_contribute | How To Contribute]]" page anywhere that people can easily find. [[User:Zerrakhi|Zerrakhi]] 19:58, 20 October 2010 (EDT)<br />
<br />
: Great. Thanks for helping to spread the word. I put a "Ways to contribute" link in the sidebar. Let me know if you have any other ideas for items in the sidebar, or anything else that might improve navigation. I'm pretty busy with work right now, but I can probably find time to get at least a little done. [[User:Oscarlevin|Oscarlevin]] 20:21, 20 October 2010 (EDT)<br />
<br />
::Well, I've at least looked at all the puzzles now, and other than the comments I've posted on a couple of puzzle talk pages, I don't think I have anything more to say. I'll check back from time to time to see what's new (and, for a while, replies to my comments), but as for my main contributions I think I'm done.<br />
<br />
::Before I go, though, I'd really like to persuade at least one more person to contribute. That would be a fitting conclusion, because I'd prefer my contributions to be not an isolated event after which everything returns to normal, but part of the ongoing story of the wiki. Much more satisfying. So I'll work on that. [[User:Zerrakhi|Zerrakhi]] 00:57, 23 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User:Zerrakhi&diff=591User:Zerrakhi2010-10-22T14:54:55Z<p>Zerrakhi: </p>
<hr />
<div>== Personal info & links ==<br />
<br />
My name is Adrian Morgan. My blog is [http://outerhoard.wordpress.com/ The Outer Hoard], and I am [http://twitter.com/GoldHoarder GoldHoarder] on Twitter. Year of birth: 1977. Residence: Adelaide, South Australia. <br />
<br />
Zerrakhi is one of several names I use on the Internet, originally a character in some stories I wrote a long time ago. There's more information on the blog.<br />
<br />
== How I got involved ==<br />
<br />
Arrived at Math Puzzle Wiki via a comment Oscar left on Richard Wiseman's blog (clicked on name). Saw puzzles in need of solutions; thought it looked like fun.<br />
<br />
== Contributions ==<br />
<br />
*Contributed solutions to some puzzles, including '''[[seven orbs]]''' (via Oscar's talk page), '''[[five towns]]''', '''[[around the world]]''', and several others.<br />
*Some editing to improve grammar, clarity, etc.<br />
*Initiated [[User_talk:Oscarlevin | discussions]] that lead to creation of [[Help:Ways to contribute]] page, etc.<br />
*Started [[Help:Editing]] page.<br />
<br />
== Favourite ==<br />
<br />
My favourite puzzle on the site is [[around the world]]. It is both elegant and challenging.</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Poison&diff=590Poison2010-10-22T14:35:05Z<p>Zerrakhi: </p>
<hr />
<div>A nice number theory puzzle based on one I heard on Car Talk.<br />
<br />
==Puzzle==<br />
<br />
In the secret lab of a mad scientist, you find thirteen vials of liquid. One of the vials contains a deadly poison that will kill you instantly. If you drink all of the other twelve vials, you will gain a pleasant variety of super-powers. Luckily, there are some petri dishes which will, in one hour, determine if any poison has been placed in them. Unluckily, there are only four to use, and you only have one hour before the mad scientist returns. How can you determine which of the thirteen vials you should definitely not drink? <br />
<br />
==Help==<br />
<br />
{{Hint| You could actually find the poisoned vial even if you have up to sixteen vials total. For seventeen vials, you would need an extra petri dish.}}<br />
<br />
{{Solution | Allocate a number to each liquid (zero, one, two, three, etc).<br />
<br />
Now let each dish represent a particular position in a four-digit binary number (so one dish represents the "eights" position, one dish represents the "fours" position, etc).<br />
<br />
Place a particular liquid in a particular dish only if the number of that liquid, written in binary, has a "1" in the position represented by that dish. <br />
<br />
Then read the results from the four dishes as a binary number, which will be the number of the liquid that is the poison.}}<br />
<br />
==References==<br />
<br />
{{Car Talk}}<br />
<br />
[[Category:Number theory]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Blank_dice&diff=589Blank dice2010-10-22T13:51:38Z<p>Zerrakhi: </p>
<hr />
<div>Here is a probability brain teaser that I remember from a math for elementary ed class I tutored. <br />
<br />
==Puzzle==<br />
<br />
When you roll two six-sided fair dice, the total number of dots displayed can be anything from 2 to 12. However, some sums are more likely than others. Now suppose you have two six-sided fair dice, with no dots on any of the faces (they are blank). Armed with only a marker pen and your puzzle-solving prowess, how can you draw dots on the dice so that each possible sum will appear with equal probability?<br />
<br />
{{Hint | Some solutions are trivial, for example you could just leave all the sides blank so that the only possible sum (zero) appears with 100% probability. But thinking about the trivial solutions may well be the first step to finding better solutions in which each dice has a different number on each side.}}<br />
<br />
[[Category: Probability]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Three_digit_magic&diff=588Three digit magic2010-10-22T08:33:47Z<p>Zerrakhi: </p>
<hr />
<div>Saw this amazing trick in one of Smullyan's books.<br />
<br />
==Puzzle==<br />
<br />
"Think of any three digit number," says the magician. "Now write that number down twice. So if you thought of 123, then you would right 123123." The participant does so. "Alright, now divide your number by 7." The participant does. "I can tell you that the result is still a whole number! But wait, there's more. Divide the result by 11. And still you have a whole number. Now divide that result by 13. Still a whole number. In fact, you are back to your original three digit number." The volunteer is amazed. Should he be?<br />
<br />
{{Solution | Certainly not. Dividing by 7, 11, and 13 is the same as dividing by their product, which is 1001. Writing a three-digit number twice is the same as multiplying it by 1001 (the first three digits are multiplied by 1000 and the second three stay the same). So you're simply multiplying by a constant and then dividing by the same constant.}}<br />
<br />
==References==<br />
<br />
{{Smullyan riddle}}<br />
<br />
[[Category: Number theory]]<br />
[[Category: Magic tricks]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Probably_blackjack&diff=587Probably blackjack2010-10-22T08:09:09Z<p>Zerrakhi: Small changes to make it easier to read.</p>
<hr />
<div>This is a classic probability puzzle. I just made up the story.<br />
<br />
==Puzzle==<br />
<br />
You are playing Blackjack and are lucky enough to have a psychic parrot on your shoulder. You are dealt three hands of two cards each, all of them face down. Because you have the help of the psychic parrot, you are only allowed to turn over one of the six cards. The parrot tells you that one hand contains two aces, one contains two jacks, and one contains an ace and a jack (a blackjack). You turn over one of the cards and see that it's an ace. What is the probability that that hand is the blackjack?<br />
<br />
==References==<br />
<br />
I got the idea for this one from {{Smullyan riddle}}<br />
<br />
[[Category: Probability]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Talk:Eight_options_with_six_sides&diff=586Talk:Eight options with six sides2010-10-22T07:43:16Z<p>Zerrakhi: Created page with "== This seems too easy == Determining the <i>least</i> number of rolls he <i>will</i> have to make is so trivial I wouldn't even call it a puzzle. Determining the <i>greatest</i..."</p>
<hr />
<div>== This seems too easy ==<br />
<br />
Determining the <i>least</i> number of rolls he <i>will</i> have to make is so trivial I wouldn't even call it a puzzle. Determining the <i>greatest</i> number of rolls he <i>may</i> have to make is a slightly less trivial, but still very simple. One can optimise either of these values or both of them. Perhaps one for [[:Category:Easy puzzles]]?<br />
<br />
As far as I can tell, I'm not missing anything. [[User:Zerrakhi|Zerrakhi]] 03:43, 22 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Eight_weights&diff=585Eight weights2010-10-21T14:26:05Z<p>Zerrakhi: Fix category (based on reasonable assumption)</p>
<hr />
<div>Here is a puzzle I saw on someones problem of the week site. They had it phrased using batteries. Here is another version.<br />
<br />
==Puzzle==<br />
<br />
You come across eight stones. You know that 4 of the stones weigh the same, while the other four all weigh different amounts. How many times might you have to use a balance scale before you can know for sure that the next two stones you will weigh will be equal?<br />
<br />
==See also==<br />
<br />
*[[Six weights]]<br />
*[[Nine weights]]<br />
*[[Twelve weights]]<br />
<br />
[[Category: Optimization puzzles]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Fill_in_the_blanks&diff=584Fill in the blanks2010-10-21T14:03:34Z<p>Zerrakhi: </p>
<hr />
<div>I've seen this puzzle a number of places, most recently as an old problem of the month on [http://mathcentral.uregina.ca/mp/previous2002/ Math Central]<br />
<br />
==Puzzle==<br />
<br />
Complete the following sentence by filling in each blank with a numeral of one or more digits.<br />
<br />
In this sentence, the number of occurrences of 0 is ____, of 1 is ____, of 2 is ____, of 3 is ____, of 4 is ____, of 5 is ____, of 6 is ____, of 7 is ____ of 8 is ____, and of 9 is ____.<br />
<br />
==Help==<br />
<br />
{{Hint| Apparently, there are exactly two solutions to this.}}<br />
{{Answer| The following sentence is one solution: <br />
<br />
In this sentence, the number of occurrences of 0 is 1, of 1 is 7, of 2 is 3, of 3 is 2, of 4 is 1, of 5 is 1, of 6 is 1, of 7 is 2 of 8 is 1, and of 9 is 1.}}<br />
<br />
==References==<br />
<br />
{{Mathcentral}}<br />
<br />
[[Category: Logic puzzles]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Three_couples&diff=583Three couples2010-10-21T13:06:01Z<p>Zerrakhi: </p>
<hr />
<div>Here is a crossing puzzle similar to [[Cannibals and missionaries]]. I first saw this in Averbach and Chein.<br />
<br />
==Puzzle==<br />
<br />
Three couples are on a safari, when they come to a river. The only way for them to cross is with a small rowboat which can only accommodate two people at a time. Complicating the process is the fact that the women are all the jealous type, and refuse to leave their significant other in the presence pf another woman unless she is there as well. How can the couples cross the river without any romantic strife?<br />
<br />
==Variations==<br />
<br />
On the way back, two extra couples have joined the group. Luckily, the boat has been upgraded and can now accommodate three passengers at a time. Sadly, the wives are just as suspicious as before. How can the group cross?<br />
<br />
In another presentation of this puzzle, the couples are replaced by pairs of knights and pages. Any page left in the company of another knight, without his own knight to protect him, would either be killed or would die of fright.<br />
<br />
==See also==<br />
[[Cannibals and missionaries]]<br />
<br />
[[Farmer and the boat]]<br />
<br />
[[Four travelers]]<br />
<br />
==References==<br />
<br />
{{Averbach}}<br />
<br />
[[Category: Crossing puzzles]]<br />
[[Category: Graph theory]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Talk:Angels_and_demons&diff=582Talk:Angels and demons2010-10-21T12:48:45Z<p>Zerrakhi: Created page with "What's to stop the demon from saying "I don't know?" It's a perfectly valid lie. (A more picky objection: Are you sure they're friends?) ~~~~"</p>
<hr />
<div>What's to stop the demon from saying "I don't know?" It's a perfectly valid lie.<br />
<br />
(A more picky objection: Are you sure they're friends?) [[User:Zerrakhi|Zerrakhi]] 08:48, 21 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Multiple_locks&diff=581Multiple locks2010-10-21T01:00:54Z<p>Zerrakhi: Spelling correction</p>
<hr />
<div>==Puzzle==<br />
<br />
You belong to a club that has accumulated valuable jewelry. The treasure is kept in a chest. For security reasons, you want at least two of your 10 club members present when the chest is open. To ensure this, you put multiple locks on the chest and distribute keys to members so that no one member can open the chest, but any two members can. Each lock has a different key, but you can make several copies of the same key to distribute to the club members. What is the fewest number of locks and keys that you will need?<br />
<br />
==Help==<br />
<br />
{{Needs answer}}<br />
<br />
==References==<br />
<br />
{{Problem Solving}}<br />
<br />
[[Category: Optimization puzzles]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User:Zerrakhi&diff=580User:Zerrakhi2010-10-21T00:39:37Z<p>Zerrakhi: </p>
<hr />
<div>== Personal info & links ==<br />
<br />
My name is Adrian Morgan. My blog is [http://outerhoard.wordpress.com/ The Outer Hoard], and I am [http://twitter.com/GoldHoarder GoldHoarder] on Twitter. Year of birth: 1977. Residence: Adelaide, South Australia. <br />
<br />
Zerrakhi is one of several names I use on the Internet, originally a character in some stories I wrote a long time ago. There's more information on the blog.<br />
<br />
== How I got involved ==<br />
<br />
Arrived at Math Puzzle Wiki via a comment Oscar left on Richard Wiseman's blog (clicked on name). Saw puzzles in need of solutions; thought it looked like fun.<br />
<br />
== Contributions ==<br />
<br />
*Contributed solutions to several puzzles, including '''[[seven orbs]]''' (via Oscar's talk page), '''[[five towns]]''', '''[[around the world]]''', and some others.<br />
*Some editing to improve grammar, clarity, etc.<br />
*Initiated [[User_talk:Oscarlevin | discussions]] that lead to creation of [[Help:Ways to contribute]] page, etc.<br />
*Started [[Help:Editing]] page.<br />
<br />
== Currently ==<br />
<br />
Using the [[Special:AncientPages | Oldest Pages]] list as an index, and trying to solve puzzles that interest me.</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=572User talk:Oscarlevin2010-10-20T23:58:41Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::: Good idea. I've started doing so.<br />
<br />
:::: Don't forget to add links to pages like that from the front page so people can find them... Also, what about adding the needs_hint, needs_answer, etc templates? Is that something people can usefully do? Since the templates exist, it seems a shame not to use them consistently. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
:::: (This section later trimmed of less important topics to enhance focus on what matters. See history for details.) [[User:Zerrakhi|Zerrakhi]] 00:23, 19 October 2010 (EDT)<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)<br />
<br />
::: The reason for trying to restrict solutions is that I want to make it difficult for students to use this site to cheat. I teach a variety of college courses where I use puzzles from this site as either extra credit assignments or as the beginnings of homework problems. If a student finds the site by searching for the puzzle, I would rather they feel intimidated by registering (because maybe they think I would see their email) and try to do the problem on their own. But I do want to have solutions somewhere to help remind me how difficult the puzzle is in case I forget how to solve it.<br />
<br />
::: I envision the site to sever as a collection of good math puzzles, with perhaps a discussion on why they are mathematically interesting. These puzzles are meant to puzzled over, not answered by a web search. Thoughts? (I probably should outline this "policy" on a help page as well.) [[User:Oscarlevin|Oscarlevin]] 22:03, 13 October 2010 (EDT)<br />
<br />
:::: Not a lot of relevant thoughts, but as you probably guessed, that was largely my reason for asking - to suggest things that would be useful to mention on the help pages. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
== Something's wrong... ==<br />
<br />
All of a sudden, I find that the hint/answer/solution boxes are '''open by default''', so that I can read their contents without wanting to! What could have caused this? (Added later: I'd say your edits to Common.js messed up the NavFrame definition somehow.) [[User:Zerrakhi|Zerrakhi]] 09:55, 18 October 2010 (EDT)<br />
<br />
: Exactly right. I undid the changes, so it should be working now. Thanks for the catch. I was trying to get the sidebar to have collapsible categories (haven't figured it out yet) and must have deleted the wrong thing. [[User:Oscarlevin|Oscarlevin]] 10:05, 18 October 2010 (EDT)<br />
<br />
:: Oddly, I've re-started my browser (to clear cookies, just in case) and still having the problem. <br />
<br />
:: (You can probably guess how I figured out the approximate cause. As puzzles go, it was easier than many on the site. Entered edit mode on solution template to see how it worked, searched common.js for mention of NavFrame, hunch confirmed.)<br />
<br />
::: Working now, though. Don't understand that. [[User:Zerrakhi|Zerrakhi]] 10:23, 18 October 2010 (EDT)<br />
<br />
:::: Must have been cached in your browser. There is a message on [[mediawiki:common.js]] about having to refresh while holding down SHIFT (for firefox anyway). [[User:Oscarlevin|Oscarlevin]] 10:25, 18 October 2010 (EDT)<br />
<br />
== Link ==<br />
<br />
Just want say that I've posted a [http://outerhoard.wordpress.com/2010/10/21/interesting-stuff-late-october-2010/ link] to the Math Puzzle Wiki on my blog.<br />
<br />
Also, you still haven't put the "[[MathPuzzleWiki:How_to_contribute | How To Contribute]]" page anywhere that people can easily find. [[User:Zerrakhi|Zerrakhi]] 19:58, 20 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User:Zerrakhi&diff=570User:Zerrakhi2010-10-20T14:13:53Z<p>Zerrakhi: </p>
<hr />
<div>== Personal info & links ==<br />
<br />
My name is Adrian Morgan. My blog is [http://outerhoard.wordpress.com/ The Outer Hoard], and I am [http://twitter.com/GoldHoarder GoldHoarder] on Twitter. Year of birth: 1977. Residence: Adelaide, South Australia. <br />
<br />
Zerrakhi is one of several names I use on the Internet, originally a character in some stories I wrote a long time ago. There's more information on the blog.<br />
<br />
== How I got involved ==<br />
<br />
Arrived at Math Puzzle Wiki via a comment Oscar left on Richard Wiseman's blog (clicked on name). Saw puzzles in need of solutions; thought it looked like fun.<br />
<br />
== Contributions ==<br />
<br />
*Contributed solutions to several puzzles, including '''[[seven orbs]]''' (via Oscar's talk page), '''[[five towns]]''', '''[[around the world]]''', and some others.<br />
*Some editing to improve grammar, clarity, etc.<br />
*Initiated [[User_talk:Oscarlevin | discussions]] that lead to creation of '''[[MathPuzzleWiki:How_to_contribute | how to contribute]]''' page, etc.<br />
*Started '''[[Help:Editing]]''' page.<br />
<br />
== Currently ==<br />
<br />
Using the [[Special:AncientPages | Oldest Pages]] list as an index, and trying to solve puzzles that interest me.</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Around_the_world&diff=569Around the world2010-10-20T14:11:32Z<p>Zerrakhi: Fix minor error (partial refill means tank not full afterward) Also change westbound to western etc to avoid confusion on return journeys.</p>
<hr />
<div>A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant speed (relative to the ground) and rate of fuel consumption, and that all planes return safely to their island base?<br />
<br />
{{Solution | First, let's agree on conventions. The main journey will be westward, and all units will be in degrees. So 360 degrees is a full circle around the earth, 1 unit of fuel is enough fuel to travel one degree, and 1 unit of time is the time taken to travel one degree.<br />
<br />
One unassisted aeroplane can travel 180 degrees non-returning, or 90 degrees returning. Let's see how far we can get with two aeroplanes: a main flight and an assisting flight.<br />
<br />
:The assisting flight requires fuel for the journey out, fuel for the transfer, and fuel for the return. The best place to transfer fuel to the main flight would be when the amount of fuel available (not needed for return) is exactly the amount of empty space in the main flight's fuel tanks. That way no fuel is wasted.<br />
<br />
:Space in tank <nowiki>=</nowiki> D (distance from base, since fuel used equals distance travelled)<br />
:Transferrable fuel <nowiki>=</nowiki> 180 - 2D (starting amount minus what's used for flight out and return)<br />
<br />
:Equating these:<br />
:D <nowiki>=</nowiki> 180 - 2D<br />
:D <nowiki>=</nowiki> 60<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 60, enabling it to reach 240 degrees non-returning (the 60 degrees already travelled plus a further 180 on a full tank), or 120 degrees returning (reserve 60 units of fuel for returning the 60 degrees already travelled, which leaves 120 units of fuel for travelling a further 60 units and back).<br />
<br />
With three aeroplanes, several scenarios are possible:<br />
*Two assisting flights both transfer fuel to the main flight at different locations on the latter's outward journey, the first at D <nowiki>=</nowiki> 60, the second later.<br />
*One flight transfers fuel to the primary assisting flight at D <nowiki>=</nowiki> 60, which in turn transfers fuel to the main flight.<br />
*Two assisting flights both transfer fuel to a main returning flight at D <nowiki>=</nowiki> 60, one on its flight out, one on its return.<br />
<br />
The first two scenarios are equivalent with respect to the optimal distance for the second transfer and the total distance that the main flight can travel. I will skip the first scenario and do the maths for the second.<br />
<br />
:Secondary assisting flight transfers fuel to primary assisting flight at D <nowiki>=</nowiki> 60.<br />
<br />
:We use the same logic as before to determine the optimal time to transfer fuel from the primary assisting flight to the main flight.<br />
<br />
:Space in main flight's tank <nowiki>=</nowiki> D<br />
:Transferrable fuel from primary assisting flight <nowiki>=</nowiki> 180 - 2(D - 60) - 60 (because tank full at D <nowiki>=</nowiki> 60 but need to travel that distance again on return)<br />
:This simplifies to 240 - 2D<br />
<br />
:D <nowiki>=</nowiki> 240 - 2D<br />
:D <nowiki>=</nowiki> 80<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 80, enabling it to reach 260 degrees non-returning or 130 degrees returning.<br />
<br />
As for the third scenario, the main flight is refueled at D <nowiki>=</nowiki> 60, enabling it to reach 150 degrees returning (90 degrees further than the 60 degrees already travelled), to be refueled again at D <nowiki>=</nowiki> 60 on its return. So this scenario achieves a longer return flight than the others at the cost of necessarily being a return flight.<br />
<br />
With four aeroplanes, we can combine the second and third scenarios above. One flight refuels the primary assisting flight at D <nowiki>=</nowiki> 60, which in turn refuels the main flight. The primary assisting flight is then refuelled by a fourth flight at D <nowiki>=</nowiki> 60 on its return.<br />
<br />
:Space in main flight's tank <nowiki>=</nowiki> D<br />
:Transferrable fuel from primary assisting flight <nowiki>=</nowiki> 180 - 2(D - 60) (this time we don't have to reserve the 60 units of fuel)<br />
:This simplifies to 300 - 2D<br />
<br />
:D <nowiki>=</nowiki> 300 - 2D<br />
:D <nowiki>=</nowiki> 100<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 100, enabling it to reach 280 degrees non-returning or 140 degrees returning.<br />
<br />
Now, 280 is only 80 short of 360 (a complete circumnavigation). Using two aeroplanes, one refueling the other, we can transport 80 units of transferable fuel 80 degrees from base. The second aeroplane, having been refueled at D <nowiki>=</nowiki> 60, travels a further 20 degrees out and must reserve enough fuel for 80 degrees return, leaving 80 units of fuel for the transfer.<br />
<br />
We can now piece together a solution. Three assisting flights travel west to enable the main flight to reach D <nowiki>=</nowiki> 280, and two more assisting flights travel east to enable the main flight to get from D <nowiki>=</nowiki> 280 to home.<br />
<br />
A more detailed itinery follows. The times (T) are easy to calculate given that the time is always equal to the location of the main flight, and that all flights travel at the same speed.<br />
<br />
*At T<nowiki>=</nowiki>0, main flight, western primary assisting flight, and 1st western secondary assisting flight begin westward journey.<br />
*At T<nowiki>=</nowiki>60, 1st western secondary assisting flight refills western primary assisting flight at D <nowiki>=</nowiki> 60.<br />
*At T<nowiki>=</nowiki>80, 2nd western secondary assisting flight begins westward journey.<br />
*At T<nowiki>=</nowiki>100, western primary assisting flight refills main flight at D <nowiki>=</nowiki> 100, giving it enough fuel to reach D <nowiki>=</nowiki> 280.<br />
*At T<nowiki>=</nowiki>120, 1st western secondary assisting flight returns to base.<br />
*At T<nowiki>=</nowiki>140, 2nd western secondary assisting flight partially refills returning western primary assisting flight at D <nowiki>=</nowiki> 60.<br />
*At T<nowiki>=</nowiki>200, western primary assisting flight and 2nd western secondary assisting flight return to base.<br />
*At T<nowiki>=</nowiki>200, eastern primary assisting flight and eastern secondary assisting flight begin eastward journey.<br />
*At T<nowiki>=</nowiki>260, eastern secondary assisting flight refills eastern primary assisting flight at D <nowiki>=</nowiki> 300 (60 from 360).<br />
*At T<nowiki>=</nowiki>280, eastern primary assisting flight partially refills main flight at D <nowiki>=</nowiki> 280 (80 from 360).<br />
*At T<nowiki>=</nowiki>320, eastern secondary assisting flight returns to base.<br />
*At T<nowiki>=</nowiki>360, main flight and eastern primary assisting flight return to base.<br />
<br />
This involves a total of six flights: one main, five returning. But by the time the eastern assisting flights have to depart, the western assisting flights have returned to the island. So the same aeroplanes can be re-used for the eastern assisting flights, reducing the number of aeroplanes needed by two.<br />
<br />
Therefore the puzzle can be solved with a total of four aeroplanes, including the main one. But certainly no fewer. Nothing is wasted in the above scenario, which confirms that it is the optimal solution. }}<br />
<br />
[[Category: Optimization puzzles]]<br />
[[Category: Geometry]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User:Zerrakhi&diff=568User:Zerrakhi2010-10-20T12:38:48Z<p>Zerrakhi: </p>
<hr />
<div>== Personal info & links ==<br />
<br />
My name is Adrian Morgan. My blog is [http://outerhoard.wordpress.com/ The Outer Hoard], and I am [http://twitter.com/GoldHoarder GoldHoarder] on Twitter. Year of birth: 1977. Residence: Adelaide, South Australia. <br />
<br />
Zerrakhi is one of several names I use on the Internet, originally a character in some stories I wrote a long time ago. There's more information on the blog.<br />
<br />
== How I got involved ==<br />
<br />
Arrived at Math Puzzle Wiki via a comment Oscar left on Richard Wiseman's blog (clicked on name). Saw puzzles in need of solutions; thought it looked like fun.<br />
<br />
== Contributions ==<br />
<br />
*Contributed solutions to several puzzles, including '''[[seven orbs]]''' (via Oscar's talk page), '''[[five towns]]''', '''[[around the world]]''', and some others.<br />
*Some editing to improve grammar, clarity, etc.<br />
*Initiated [[User_talk:Oscarlevin | discussions]] that lead to creation of '''[[MathPuzzleWiki:How_to_contribute | how to contribute]]''' page, etc.<br />
*Started '''[[Help:Editing]]''' page.<br />
<br />
== Currently ==<br />
<br />
Using the '''[[Special:AncientPages | Oldest Pages]]''' list as an index, and trying to solve puzzles that interest me.</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Around_the_world&diff=567Around the world2010-10-20T12:36:52Z<p>Zerrakhi: </p>
<hr />
<div>A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant speed (relative to the ground) and rate of fuel consumption, and that all planes return safely to their island base?<br />
<br />
{{Solution | First, let's agree on conventions. The main journey will be westward, and all units will be in degrees. So 360 degrees is a full circle around the earth, 1 unit of fuel is enough fuel to travel one degree, and 1 unit of time is the time taken to travel one degree.<br />
<br />
One unassisted aeroplane can travel 180 degrees non-returning, or 90 degrees returning. Let's see how far we can get with two aeroplanes: a main flight and an assisting flight.<br />
<br />
:The assisting flight requires fuel for the journey out, fuel for the transfer, and fuel for the return. The best place to transfer fuel to the main flight would be when the amount of fuel available (not needed for return) is exactly the amount of empty space in the main flight's fuel tanks. That way no fuel is wasted.<br />
<br />
:Space in tank <nowiki>=</nowiki> D (distance from base, since fuel used equals distance travelled)<br />
:Transferrable fuel <nowiki>=</nowiki> 180 - 2D (starting amount minus what's used for flight out and return)<br />
<br />
:Equating these:<br />
:D <nowiki>=</nowiki> 180 - 2D<br />
:D <nowiki>=</nowiki> 60<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 60, enabling it to reach 240 degrees non-returning (the 60 degrees already travelled plus a further 180 on a full tank), or 120 degrees returning (reserve 60 units of fuel for returning the 60 degrees already travelled, which leaves 120 units of fuel for travelling a further 60 units and back).<br />
<br />
With three aeroplanes, several scenarios are possible:<br />
*Two assisting flights both transfer fuel to the main flight at different locations on the latter's outward journey, the first at D <nowiki>=</nowiki> 60, the second later.<br />
*One flight transfers fuel to the primary assisting flight at D <nowiki>=</nowiki> 60, which in turn transfers fuel to the main flight.<br />
*Two assisting flights both transfer fuel to a main returning flight at D <nowiki>=</nowiki> 60, one on its flight out, one on its return.<br />
<br />
The first two scenarios are equivalent with respect to the optimal distance for the second transfer and the total distance that the main flight can travel. I will skip the first scenario and do the maths for the second.<br />
<br />
:Secondary assisting flight transfers fuel to primary assisting flight at D <nowiki>=</nowiki> 60.<br />
<br />
:We use the same logic as before to determine the optimal time to transfer fuel from the primary assisting flight to the main flight.<br />
<br />
:Space in main flight's tank <nowiki>=</nowiki> D<br />
:Transferrable fuel from primary assisting flight <nowiki>=</nowiki> 180 - 2(D - 60) - 60 (because tank full at D <nowiki>=</nowiki> 60 but need to travel that distance again on return)<br />
:This simplifies to 240 - 2D<br />
<br />
:D <nowiki>=</nowiki> 240 - 2D<br />
:D <nowiki>=</nowiki> 80<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 80, enabling it to reach 260 degrees non-returning or 130 degrees returning.<br />
<br />
As for the third scenario, the main flight is refueled at D <nowiki>=</nowiki> 60, enabling it to reach 150 degrees returning (90 degrees further than the 60 degrees already travelled), to be refueled again at D <nowiki>=</nowiki> 60 on its return. So this scenario achieves a longer return flight than the others at the cost of necessarily being a return flight.<br />
<br />
With four aeroplanes, we can combine the second and third scenarios above. One flight refuels the primary assisting flight at D <nowiki>=</nowiki> 60, which in turn refuels the main flight. The primary assisting flight is then refuelled by a fourth flight at D <nowiki>=</nowiki> 60 on its return.<br />
<br />
:Space in main flight's tank <nowiki>=</nowiki> D<br />
:Transferrable fuel from primary assisting flight <nowiki>=</nowiki> 180 - 2(D - 60) (this time we don't have to reserve the 60 units of fuel)<br />
:This simplifies to 300 - 2D<br />
<br />
:D <nowiki>=</nowiki> 300 - 2D<br />
:D <nowiki>=</nowiki> 100<br />
<br />
:So the main flight is refueled at D <nowiki>=</nowiki> 100, enabling it to reach 280 degrees non-returning or 140 degrees returning.<br />
<br />
Now, 280 is only 80 short of 360 (a complete circumnavigation). Using two aeroplanes, one refueling the other, we can transport 80 units of transferable fuel 80 degrees from base. The second aeroplane, having been refueled at D <nowiki>=</nowiki> 60, travels a further 20 degrees out and must reserve enough fuel for 80 degrees return, leaving 80 units of fuel for the transfer.<br />
<br />
We can now piece together a solution. Three assisting flights travel west to enable the main flight to reach D <nowiki>=</nowiki> 280, and two more assisting flights travel east to enable the main flight to get from D <nowiki>=</nowiki> 280 to home.<br />
<br />
A more detailed itinery follows. The times (T) are easy to calculate given that the time is always equal to the location of the main flight, and that all flights travel at the same speed.<br />
<br />
*At T<nowiki>=</nowiki>0, main flight, westbound primary assisting flight, and 1st westbound secondary assisting flight begin westward journey.<br />
*At T<nowiki>=</nowiki>60, 1st westbound secondary assisting flight refills westbound primary assisting flight at D <nowiki>=</nowiki> 60.<br />
*At T<nowiki>=</nowiki>80, 2nd westbound secondary assisting flight begins westward journey.<br />
*At T<nowiki>=</nowiki>100, westbound primary assisting flight refills main flight at D <nowiki>=</nowiki> 100, giving it enough fuel to reach D <nowiki>=</nowiki> 280.<br />
*At T<nowiki>=</nowiki>120, 1st westbound secondary assisting flight returns to base.<br />
*At T<nowiki>=</nowiki>140, 2nd westbound secondary assisting flight refills returning westbound primary assisting flight at D <nowiki>=</nowiki> 60.<br />
*At T<nowiki>=</nowiki>200, westbound primary assisting flight and 2nd westbound secondary assisting flight return to base.<br />
*At T<nowiki>=</nowiki>200, eastbound primary assisting flight and eastbound secondary assisting flight begin eastward journey.<br />
*At T<nowiki>=</nowiki>260, eastbound secondary assisting flight refills eastbound primary assisting flight at D <nowiki>=</nowiki> 300 (60 from 360).<br />
*At T<nowiki>=</nowiki>280, eastbound primary assisting flight partially refills main flight at D <nowiki>=</nowiki> 280 (80 from 360).<br />
*At T<nowiki>=</nowiki>320, eastbound secondary assisting flight returns to base.<br />
*At T<nowiki>=</nowiki>360, main flight and eastbound primary assisting flight return to base.<br />
<br />
This involves a total of six flights: one main, five returning. But by the time the eastbound assisting flights have to depart, the westbound assisting flights have returned to the island. So the same aeroplanes can be re-used for the eastbound assisting flights, reducing the number of aeroplanes needed by two.<br />
<br />
Therefore the puzzle can be solved with a total of four aeroplanes, including the main one. But certainly no fewer. Nothing is wasted in the above scenario, which confirms that it is the optimal solution. }}<br />
<br />
[[Category: Optimization puzzles]]<br />
[[Category: Geometry]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User:Zerrakhi&diff=566User:Zerrakhi2010-10-19T08:37:18Z<p>Zerrakhi: </p>
<hr />
<div>== Personal info & links ==<br />
<br />
My name is Adrian Morgan. My blog is [http://outerhoard.wordpress.com/ The Outer Hoard], and I am [http://twitter.com/GoldHoarder GoldHoarder] on Twitter. Year of birth: 1977. Residence: Adelaide, South Australia. <br />
<br />
Zerrakhi is one of several names I use on the Internet, originally a character in some stories I wrote a long time ago. There's more information on the blog.<br />
<br />
== How I got involved ==<br />
<br />
Arrived at Math Puzzle Wiki via a comment Oscar left on Richard Wiseman's blog (clicked on name). Saw puzzles in need of solutions; thought it looked like fun.<br />
<br />
== Contributions ==<br />
<br />
*Contributed solutions to several puzzles, including '''[[seven orbs]]''' (via Oscar's talk page), '''[[five towns]]''', etc.<br />
*Some editing to improve grammar, clarity, etc.<br />
*Initiated [[User_talk:Oscarlevin | discussions]] that lead to creation of '''[[MathPuzzleWiki:How_to_contribute | how to contribute]]''' page, etc.<br />
*Started '''[[Help:Editing]]''' page.<br />
<br />
== Currently ==<br />
<br />
Using the '''[[Special:AncientPages | Oldest Pages]]''' list as an index, and trying to solve puzzles that interest me. (At most recent update, '''[[around the world]]''' is next on my list of puzzles to ponder.)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Around_the_world&diff=565Around the world2010-10-19T04:41:04Z<p>Zerrakhi: Get rid of aviation jargon ("ground speed")</p>
<hr />
<div>A group of airplanes is based on a small island. The tank of each plane holds just enough fuel to take it halfway around the world. Any desired amount of fuel can be transferred from the tank of one plane to the tank of another while the planes are in flight. The only source of fuel is on the island, and it is assumed that there is no time lost in refueling either in the air or on the ground. What is the smallest number of planes that will ensure the flight of one plane around the world on a great circle, assuming that the planes have the same constant speed (relative to the ground) and rate of fuel consumption, and that all planes return safely to their island base?<br />
<br />
[[Category: Optimization puzzles]]<br />
[[Category: Geometry]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=564User talk:Oscarlevin2010-10-19T04:23:18Z<p>Zerrakhi: Trimmed some less important topics from "More questions" section, to avoid cluttering of priority issues.</p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::: Good idea. I've started doing so.<br />
<br />
:::: Don't forget to add links to pages like that from the front page so people can find them... Also, what about adding the needs_hint, needs_answer, etc templates? Is that something people can usefully do? Since the templates exist, it seems a shame not to use them consistently. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
:::: (This section later trimmed of less important topics to enhance focus on what matters. See history for details.) [[User:Zerrakhi|Zerrakhi]] 00:23, 19 October 2010 (EDT)<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)<br />
<br />
::: The reason for trying to restrict solutions is that I want to make it difficult for students to use this site to cheat. I teach a variety of college courses where I use puzzles from this site as either extra credit assignments or as the beginnings of homework problems. If a student finds the site by searching for the puzzle, I would rather they feel intimidated by registering (because maybe they think I would see their email) and try to do the problem on their own. But I do want to have solutions somewhere to help remind me how difficult the puzzle is in case I forget how to solve it.<br />
<br />
::: I envision the site to sever as a collection of good math puzzles, with perhaps a discussion on why they are mathematically interesting. These puzzles are meant to puzzled over, not answered by a web search. Thoughts? (I probably should outline this "policy" on a help page as well.) [[User:Oscarlevin|Oscarlevin]] 22:03, 13 October 2010 (EDT)<br />
<br />
:::: Not a lot of relevant thoughts, but as you probably guessed, that was largely my reason for asking - to suggest things that would be useful to mention on the help pages. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
== Something's wrong... ==<br />
<br />
All of a sudden, I find that the hint/answer/solution boxes are '''open by default''', so that I can read their contents without wanting to! What could have caused this? (Added later: I'd say your edits to Common.js messed up the NavFrame definition somehow.) [[User:Zerrakhi|Zerrakhi]] 09:55, 18 October 2010 (EDT)<br />
<br />
: Exactly right. I undid the changes, so it should be working now. Thanks for the catch. I was trying to get the sidebar to have collapsible categories (haven't figured it out yet) and must have deleted the wrong thing. [[User:Oscarlevin|Oscarlevin]] 10:05, 18 October 2010 (EDT)<br />
<br />
:: Oddly, I've re-started my browser (to clear cookies, just in case) and still having the problem. <br />
<br />
:: (You can probably guess how I figured out the approximate cause. As puzzles go, it was easier than many on the site. Entered edit mode on solution template to see how it worked, searched common.js for mention of NavFrame, hunch confirmed.)<br />
<br />
::: Working now, though. Don't understand that. [[User:Zerrakhi|Zerrakhi]] 10:23, 18 October 2010 (EDT)<br />
<br />
:::: Must have been cached in your browser. There is a message on [[mediawiki:common.js]] about having to refresh while holding down SHIFT (for firefox anyway). [[User:Oscarlevin|Oscarlevin]] 10:25, 18 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User:Zerrakhi&diff=563User:Zerrakhi2010-10-19T04:05:23Z<p>Zerrakhi: Created page with '== Personal info & links == My name is Adrian Morgan. My blog is [http://outerhoard.wordpress.com/ The Outer Hoard], and I am [http://twitter.com/GoldHoarder GoldHoarder] on Twi...'</p>
<hr />
<div>== Personal info & links ==<br />
<br />
My name is Adrian Morgan. My blog is [http://outerhoard.wordpress.com/ The Outer Hoard], and I am [http://twitter.com/GoldHoarder GoldHoarder] on Twitter. Year of birth: 1977. Residence: Adelaide, South Australia. <br />
<br />
Zerrakhi is one of several names I use on the Internet, originally a character in some stories I wrote a long time ago. There's more information on the blog.<br />
<br />
== How I got involved ==<br />
<br />
Arrived at MathWiki via a comment Oscar left on Richard Wiseman's blog (clicked on name). Saw puzzles in need of solutions; thought it looked like fun.<br />
<br />
== Contributions ==<br />
<br />
*Contributed solutions to several puzzles, including '''[[seven orbs]]''' (via Oscar's talk page), '''[[five towns]]''', etc.<br />
*Some editing to improve grammar, clarity, etc.<br />
*Initiated [[User_talk:Oscarlevin | discussions]] that lead to creation of '''[[MathPuzzleWiki:How_to_contribute | how to contribute]]''' page, etc.<br />
*Started '''[[Help:Editing]]''' page.<br />
<br />
== Currently ==<br />
<br />
Using the '''[[Special:AncientPages | Oldest Pages]]''' list as an index, and trying to solve puzzles that interest me. (At most recent update, '''[[around the world]]''' is next on my list of puzzles to ponder.)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=561User talk:Oscarlevin2010-10-18T14:23:26Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::: Good idea. I've started doing so.<br />
<br />
:::: Don't forget to add links to pages like that from the front page so people can find them... Also, what about adding the needs_hint, needs_answer, etc templates? Is that something people can usefully do? Since the templates exist, it seems a shame not to use them consistently.<br />
<br />
:::: I'll give you a link from my blog sometime this week. (Not that it will do any good: very few people read it.) As for self promotion, nobody will mind if you mention it in a postscript next time you leave a comment on Richard Wiseman's blog; he's never discouraged people from using it as a giant conversation forum and it will reach a wider audience than I ever could. <br />
<br />
:::: Incidentally, I knew the lightbulb puzzle because it's been featured on Wiseman's blog in the past. Also, some time ago I nominated a puzzle that he has confirmed he will use in some future installment. If you want a spoiler, it's the...<br />
<br />
{{answer | Knights and Pages puzzle, a version of which can be found at http://www.outofworkpoet.com/three-knights-and-three-pages . In the book I got it from, it comes in three parts: (1) as in the link, (2) proving there is no solution with four pairs of characters. (3) solving with four pairs of characters plus an island.}}<br />
<br />
:::: I'll have plenty to say about that when it gets to the top of the queue, but not before. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)<br />
<br />
::: The reason for trying to restrict solutions is that I want to make it difficult for students to use this site to cheat. I teach a variety of college courses where I use puzzles from this site as either extra credit assignments or as the beginnings of homework problems. If a student finds the site by searching for the puzzle, I would rather they feel intimidated by registering (because maybe they think I would see their email) and try to do the problem on their own. But I do want to have solutions somewhere to help remind me how difficult the puzzle is in case I forget how to solve it.<br />
<br />
::: I envision the site to sever as a collection of good math puzzles, with perhaps a discussion on why they are mathematically interesting. These puzzles are meant to puzzled over, not answered by a web search. Thoughts? (I probably should outline this "policy" on a help page as well.) [[User:Oscarlevin|Oscarlevin]] 22:03, 13 October 2010 (EDT)<br />
<br />
:::: Not a lot of relevant thoughts, but as you probably guessed, that was largely my reason for asking - to suggest things that would be useful to mention on the help pages. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
== Something's wrong... ==<br />
<br />
All of a sudden, I find that the hint/answer/solution boxes are '''open by default''', so that I can read their contents without wanting to! What could have caused this? (Added later: I'd say your edits to Common.js messed up the NavFrame definition somehow.) [[User:Zerrakhi|Zerrakhi]] 09:55, 18 October 2010 (EDT)<br />
<br />
: Exactly right. I undid the changes, so it should be working now. Thanks for the catch. I was trying to get the sidebar to have collapsible categories (haven't figured it out yet) and must have deleted the wrong thing. [[User:Oscarlevin|Oscarlevin]] 10:05, 18 October 2010 (EDT)<br />
<br />
:: Oddly, I've re-started my browser (to clear cookies, just in case) and still having the problem. <br />
<br />
:: (You can probably guess how I figured out the approximate cause. As puzzles go, it was easier than many on the site. Entered edit mode on solution template to see how it worked, searched common.js for mention of NavFrame, hunch confirmed.)<br />
<br />
::: Working now, though. Don't understand that. [[User:Zerrakhi|Zerrakhi]] 10:23, 18 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=560User talk:Oscarlevin2010-10-18T14:16:23Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::: Good idea. I've started doing so.<br />
<br />
:::: Don't forget to add links to pages like that from the front page so people can find them... Also, what about adding the needs_hint, needs_answer, etc templates? Is that something people can usefully do? Since the templates exist, it seems a shame not to use them consistently.<br />
<br />
:::: I'll give you a link from my blog sometime this week. (Not that it will do any good: very few people read it.) As for self promotion, nobody will mind if you mention it in a postscript next time you leave a comment on Richard Wiseman's blog; he's never discouraged people from using it as a giant conversation forum and it will reach a wider audience than I ever could. <br />
<br />
:::: Incidentally, I knew the lightbulb puzzle because it's been featured on Wiseman's blog in the past. Also, some time ago I nominated a puzzle that he has confirmed he will use in some future installment. If you want a spoiler, it's the...<br />
<br />
{{answer | Knights and Pages puzzle, a version of which can be found at http://www.outofworkpoet.com/three-knights-and-three-pages . In the book I got it from, it comes in three parts: (1) as in the link, (2) proving there is no solution with four pairs of characters. (3) solving with four pairs of characters plus an island.}}<br />
<br />
:::: I'll have plenty to say about that when it gets to the top of the queue, but not before. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)<br />
<br />
::: The reason for trying to restrict solutions is that I want to make it difficult for students to use this site to cheat. I teach a variety of college courses where I use puzzles from this site as either extra credit assignments or as the beginnings of homework problems. If a student finds the site by searching for the puzzle, I would rather they feel intimidated by registering (because maybe they think I would see their email) and try to do the problem on their own. But I do want to have solutions somewhere to help remind me how difficult the puzzle is in case I forget how to solve it.<br />
<br />
::: I envision the site to sever as a collection of good math puzzles, with perhaps a discussion on why they are mathematically interesting. These puzzles are meant to puzzled over, not answered by a web search. Thoughts? (I probably should outline this "policy" on a help page as well.) [[User:Oscarlevin|Oscarlevin]] 22:03, 13 October 2010 (EDT)<br />
<br />
:::: Not a lot of relevant thoughts, but as you probably guessed, that was largely my reason for asking - to suggest things that would be useful to mention on the help pages. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
== Something's wrong... ==<br />
<br />
All of a sudden, I find that the hint/answer/solution boxes are '''open by default''', so that I can read their contents without wanting to! What could have caused this? (Added later: I'd say your edits to Common.js messed up the NavFrame definition somehow.) [[User:Zerrakhi|Zerrakhi]] 09:55, 18 October 2010 (EDT)<br />
<br />
: Exactly right. I undid the changes, so it should be working now. Thanks for the catch. I was trying to get the sidebar to have collapsible categories (haven't figured it out yet) and must have deleted the wrong thing. [[User:Oscarlevin|Oscarlevin]] 10:05, 18 October 2010 (EDT)<br />
<br />
:: Oddly, I've re-started my browser (to clear cookies, just in case) and still having the problem. <br />
<br />
:: (You can probably guess how I figured out the approximate cause. As puzzles go, it was easier than many on the site. Entered edit mode on solution template to see how it worked, searched common.js for mention of NavFrame, hunch confirmed.)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=558User talk:Oscarlevin2010-10-18T14:02:54Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::: Good idea. I've started doing so.<br />
<br />
:::: Don't forget to add links to pages like that from the front page so people can find them... Also, what about adding the needs_hint, needs_answer, etc templates? Is that something people can usefully do? Since the templates exist, it seems a shame not to use them consistently.<br />
<br />
:::: I'll give you a link from my blog sometime this week. (Not that it will do any good: very few people read it.) As for self promotion, nobody will mind if you mention it in a postscript next time you leave a comment on Richard Wiseman's blog; he's never discouraged people from using it as a giant conversation forum and it will reach a wider audience than I ever could. <br />
<br />
:::: Incidentally, I knew the lightbulb puzzle because it's been featured on Wiseman's blog in the past. Also, some time ago I nominated a puzzle that he has confirmed he will use in some future installment. If you want a spoiler, it's the...<br />
<br />
{{answer | Knights and Pages puzzle, a version of which can be found at http://www.outofworkpoet.com/three-knights-and-three-pages . In the book I got it from, it comes in three parts: (1) as in the link, (2) proving there is no solution with four pairs of characters. (3) solving with four pairs of characters plus an island.}}<br />
<br />
:::: I'll have plenty to say about that when it gets to the top of the queue, but not before. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)<br />
<br />
::: The reason for trying to restrict solutions is that I want to make it difficult for students to use this site to cheat. I teach a variety of college courses where I use puzzles from this site as either extra credit assignments or as the beginnings of homework problems. If a student finds the site by searching for the puzzle, I would rather they feel intimidated by registering (because maybe they think I would see their email) and try to do the problem on their own. But I do want to have solutions somewhere to help remind me how difficult the puzzle is in case I forget how to solve it.<br />
<br />
::: I envision the site to sever as a collection of good math puzzles, with perhaps a discussion on why they are mathematically interesting. These puzzles are meant to puzzled over, not answered by a web search. Thoughts? (I probably should outline this "policy" on a help page as well.) [[User:Oscarlevin|Oscarlevin]] 22:03, 13 October 2010 (EDT)<br />
<br />
:::: Not a lot of relevant thoughts, but as you probably guessed, that was largely my reason for asking - to suggest things that would be useful to mention on the help pages. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
== Something's wrong... ==<br />
<br />
All of a sudden, I find that the hint/answer/solution boxes are '''open by default''', so that I can read their contents without wanting to! What could have caused this? (Added later: I'd say your edits to Common.js messed up the NavFrame definition somehow.) [[User:Zerrakhi|Zerrakhi]] 09:55, 18 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=555User talk:Oscarlevin2010-10-18T13:55:59Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::: Good idea. I've started doing so.<br />
<br />
:::: Don't forget to add links to pages like that from the front page so people can find them... Also, what about adding the needs_hint, needs_answer, etc templates? Is that something people can usefully do? Since the templates exist, it seems a shame not to use them consistently.<br />
<br />
:::: I'll give you a link from my blog sometime this week. (Not that it will do any good: very few people read it.) As for self promotion, nobody will mind if you mention it in a postscript next time you leave a comment on Richard Wiseman's blog; he's never discouraged people from using it as a giant conversation forum and it will reach a wider audience than I ever could. <br />
<br />
:::: Incidentally, I knew the lightbulb puzzle because it's been featured on Wiseman's blog in the past. Also, some time ago I nominated a puzzle that he has confirmed he will use in some future installment. If you want a spoiler, it's the...<br />
<br />
{{answer | Knights and Pages puzzle, a version of which can be found at http://www.outofworkpoet.com/three-knights-and-three-pages . In the book I got it from, it comes in three parts: (1) as in the link, (2) proving there is no solution with four pairs of characters. (3) solving with four pairs of characters plus an island.}}<br />
<br />
:::: I'll have plenty to say about that when it gets to the top of the queue, but not before. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)<br />
<br />
::: The reason for trying to restrict solutions is that I want to make it difficult for students to use this site to cheat. I teach a variety of college courses where I use puzzles from this site as either extra credit assignments or as the beginnings of homework problems. If a student finds the site by searching for the puzzle, I would rather they feel intimidated by registering (because maybe they think I would see their email) and try to do the problem on their own. But I do want to have solutions somewhere to help remind me how difficult the puzzle is in case I forget how to solve it.<br />
<br />
::: I envision the site to sever as a collection of good math puzzles, with perhaps a discussion on why they are mathematically interesting. These puzzles are meant to puzzled over, not answered by a web search. Thoughts? (I probably should outline this "policy" on a help page as well.) [[User:Oscarlevin|Oscarlevin]] 22:03, 13 October 2010 (EDT)<br />
<br />
:::: Not a lot of relevant thoughts, but as you probably guessed, that was largely my reason for asking - to suggest things that would be useful to mention on the help pages. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
== Something's wrong... ==<br />
<br />
All of a sudden, I find that the hint/answer/solution boxes are '''open by default''', so that I can read their contents without wanting to! What could have caused this? [[User:Zerrakhi|Zerrakhi]] 09:55, 18 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=GO_ON_BOB&diff=554GO ON BOB2010-10-18T13:35:17Z<p>Zerrakhi: </p>
<hr />
<div>Solve this cryptarithmetic (each letter represents a unique digit 0 to 9). GO + ON = BOB, given that BOB is divisible by G.<br />
<br />
{{solution | The sum of two 2-digit numbers has to be less than 200 (because 99+99<nowiki>=</nowiki>198), so B < 2.<br />
<br />
B is not 0, because that would require the sum of two 2-digit numbers to be less than one of those 2-digit numbers (specifically, "BOB" would be "ON" rounded down to a lower multiple of ten).<br />
<br />
Hence B <nowiki>=</nowiki> 1. Which means "BOB" is one of 101, 121, 131, etc. (Not 111, because each letter represents a unique digit.)<br />
<br />
Knowing that "BOB" is divisible by G, we can look up 1-digit factors of candidates for "BOB". (In the old days, mathematicians would have had tables for this. We have the Internet.)<br />
<br />
141 has factor 3; 161 has factor 7; 171 has factors 3, 9. No other candidates for "BOB" have 1-digit factors other than 1.<br />
<br />
Trying the candidates one at a time:<br />
*34 + 4_ <nowiki>=</nowiki> 141 (obviously not, as 34+49 is only 83)<br />
*76 + 6_ <nowiki>=</nowiki> 161 (obviously not, as 76+69 is only 145)<br />
*97 + 7_ <nowiki>=</nowiki> 171 (this is the only candidate where the sum is in the right range)<br />
<br />
171 - 97 <nowiki>=</nowiki> 74, which gives 97 + 74 <nowiki>=</nowiki> 171 as the equation represented by the words. }}<br />
<br />
[[Category: Cryptarithmetic puzzles]]<br />
[[Category: Number theory]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Poison&diff=547Poison2010-10-18T00:35:26Z<p>Zerrakhi: Got rid of word 'administered'. I don't think you can administer to an object (such as a petri dish) can you? Only to a person, I'd have thought.</p>
<hr />
<div>A nice number theory puzzle based on one I heard on Car Talk.<br />
<br />
==Puzzle==<br />
<br />
In the secret lab of a mad scientist, you find thirteen vials of liquid. One of the vials contains a deadly poison that will kill you instantly. If you drink all of the other twelve vials, you will gain a pleasant variety of super-powers. Luckily, there are some petri dishes which will, in one hour, determine if any poison has been placed in them. Unluckily, there are only four to use, and you only have one hour before the mad scientist returns. How can you determine which of the thirteen vials you should definitely not drink? <br />
<br />
==Help==<br />
<br />
{{Hint| You could actually find the poisoned vial even if you have up to sixteen vials total. For seventeen vials, you would need an extra petri dish.}}<br />
<br />
==References==<br />
<br />
{{Car Talk}}<br />
<br />
[[Category:Number theory]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Five_towns&diff=546Five towns2010-10-17T23:25:51Z<p>Zerrakhi: </p>
<hr />
<div>A puzzle similar to the [[Houses and utilities]] puzzle.<br />
<br />
==Puzzle==<br />
<br />
Back in the days of yore, five small towns decided they wanted to build roads connecting each pair of towns. While the towns had plenty of money to build roads as long as they wished, it was very important that the roads not intersect with each other (as stop signs had not yet been invented). Also, tunnels and bridges were not allowed. Is it possible for each of these town to build a road to each of the four other towns without creating any intersections?<br />
<br />
{{solution | Here's one way to prove that it's not possible.<br />
<br />
Label the towns 1 to 5. If there's a solution, then for every possible ordering of the five towns, there is a circuit route that visits all five of them in that order and returns to the first, with no backtracking.<br />
<br />
In particular, there must be a circuit route that visits the towns in the order 1, 2, 3, 4, 5, 1 (call this Circuit A) and a circuit route that visits the towns in the order 1, 3, 5, 2, 4, 1 (call this Circuit B).<br />
<br />
You can quickly show that Circuit B cannot be entirely inside Circuit A, nor can it be entirely outside Circuit A. Therefore, there must be at least one town (call it N) where Circuit B crosses from being outside Circuit A to being inside Circuit A.<br />
<br />
But, as can be seen by sketching a diagram, this necessarily seperates town N-1 from town N+1 so that no road is possible between the two. }}<br />
<br />
{{Needs math}}<br />
<br />
[[Category:Graph theory]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Five_towns&diff=545Five towns2010-10-17T23:25:01Z<p>Zerrakhi: </p>
<hr />
<div>A puzzle similar to the [[Houses and utilities]] puzzle.<br />
<br />
==Puzzle==<br />
<br />
Back in the days of yore, five small towns decided they wanted to build roads connecting each pair of towns. While the towns had plenty of money to build roads as long as they wished, it was very important that the roads not intersect with each other (as stop signs had not yet been invented). Also, tunnels and bridges were not allowed. Is it possible for each of these town to build a road to each of the four other towns without creating any intersections?<br />
<br />
{{solution | Here's one way to prove that it's not possible.<br />
<br />
Label the towns 1 to 5. If there's a solution, then for every possible ordering of the five towns, there is a circuit route that visits all five of them in that order and returns to the first, with no backtracking.<br />
<br />
In particular, there must be a circuit route that visits the towns in the order 1, 2, 3, 4, 5, 1 (call this Circuit A) and a circuit route that visits the towns in the order 1, 3, 5, 2, 4, 1.<br />
<br />
You can quickly show that Circuit B cannot be entirely inside Circuit A, nor can it be entirely outside Circuit A. Therefore, there must be at least one town (call it N) where Circuit B crosses from being outside Circuit A to being inside Circuit A.<br />
<br />
But, as can be seen by sketching a diagram, this necessarily seperates town N-1 from town N+1 so that no road is possible between the two. }}<br />
<br />
{{Needs math}}<br />
<br />
[[Category:Graph theory]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Two_guards&diff=544Two guards2010-10-17T15:12:06Z<p>Zerrakhi: Alternative answer (Source: The Chicken from Minsk by Yuri B Chernyak & Robert M Rose)</p>
<hr />
<div>Here is the classic puzzle of the two guards, one a liar and one honest. As seen in Labyrinth, Doctor Who, and elsewhere.<br />
<br />
==Puzzle==<br />
<br />
You are in a dungeon (trying to get out, of course) and you encounter two doors with a centurion guarding each one. One guard always lies and the other<br />
always tells the truth, but you do not know which is which. You are allowed one question to determine the correct door. (The correct door leads to a beautiful princess, a kingâ€™s ransom, and the exit, while the incorrect door leads to a man-eating lion, and horrible death.) What question should you ask, and to whom, to ensure your safety?<br />
<br />
==Help==<br />
<br />
{{Hint| The guards know about each other. For example (although this does not help) if the first guard was a liar he would say that the second guard is a liar (because it is a lie to say the second guard is a liar).}}<br />
{{Hint| It is impossible (and unnecessary) to determine which guard is a truth-teller and which is a liar.}}<br />
<br />
{{Answer| Ask either guard, "If I were to ask the other guard which door leads to safety, what would he say?" Go through the ''other'' door.<br />
<br />
Alternative answer: Ask either guard "Is the truthful guard standing in front of the door that leads to death?" If the answer is no, go through that door; if yes, go through the other one.}}<br />
<br />
{{Solution| Why does this work? Let's consider the two cases: (1) the guard you ask always lies or (2) the guard you ask always tells the truth. In case 1, the other guard always tells the truth, so if you asked him which door leads to safety, he would tell you the correct door. The guard you asked must lie about that though, so he will tell you the other door. In case 2, the other guard always lies, so would tell you the wrong door, and the first guard faithfully conveys this information to you. In either case, you will be told the wrong door, so going through the other door is the correct play.<br />
<br />
The alternative answer is less well known, perhaps because it doesn't work for versions in which the guards don't stand directly in front of the doors. (Also because it's never appeared on television.) But it works on a very similar principle.}}<br />
<br />
[[Category: Liar puzzles]]<br />
[[Category: Logic]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Talk:Five_pirates&diff=543Talk:Five pirates2010-10-17T13:49:57Z<p>Zerrakhi: </p>
<hr />
<div>== Error or variation? ==<br />
<br />
Based on comparisons with other versions that I can find online, I believe "''If more than half of the pirates accept the proposal''" should read "''If at least half of the pirates accept the proposal''". <br />
<br />
This is a serious issue, as the puzzle depends critically upon what happens when exactly 50% of pirates accept a proposal. Is it an error, or a variation?<br />
<br />
: It is a variation. The [[Forty thieves]] puzzle uses "at least half" and as such is considerably easier to solve. [[User:Oscarlevin|Oscarlevin]] 13:09, 14 October 2010 (EDT)<br />
<br />
:: OK. To fill in some background here, I came up with a solution (I think), but I had to make an assumption (namely that a pirate will always let another pirate live if there's no financial gain from having them killed) which I felt wasn't in character and thus felt unsatisfied with it. So later I looked online to see if I was right and only found versions with "at least half". [[User:Zerrakhi|Zerrakhi]] 09:49, 17 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=542User talk:Oscarlevin2010-10-17T13:33:30Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::: Good idea. I've started doing so.<br />
<br />
:::: Don't forget to add links to pages like that from the front page so people can find them... Also, what about adding the needs_hint, needs_answer, etc templates? Is that something people can usefully do? Since the templates exist, it seems a shame not to use them consistently.<br />
<br />
:::: I'll give you a link from my blog sometime this week. (Not that it will do any good: very few people read it.) As for self promotion, nobody will mind if you mention it in a postscript next time you leave a comment on Richard Wiseman's blog; he's never discouraged people from using it as a giant conversation forum and it will reach a wider audience than I ever could. <br />
<br />
:::: Incidentally, I knew the lightbulb puzzle because it's been featured on Wiseman's blog in the past. Also, some time ago I nominated a puzzle that he has confirmed he will use in some future installment. If you want a spoiler, it's the...<br />
<br />
{{answer | Knights and Pages puzzle, a version of which can be found at http://www.outofworkpoet.com/three-knights-and-three-pages . In the book I got it from, it comes in three parts: (1) as in the link, (2) proving there is no solution with four pairs of characters. (3) solving with four pairs of characters plus an island.}}<br />
<br />
:::: I'll have plenty to say about that when it gets to the top of the queue, but not before. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)<br />
<br />
::: The reason for trying to restrict solutions is that I want to make it difficult for students to use this site to cheat. I teach a variety of college courses where I use puzzles from this site as either extra credit assignments or as the beginnings of homework problems. If a student finds the site by searching for the puzzle, I would rather they feel intimidated by registering (because maybe they think I would see their email) and try to do the problem on their own. But I do want to have solutions somewhere to help remind me how difficult the puzzle is in case I forget how to solve it.<br />
<br />
::: I envision the site to sever as a collection of good math puzzles, with perhaps a discussion on why they are mathematically interesting. These puzzles are meant to puzzled over, not answered by a web search. Thoughts? (I probably should outline this "policy" on a help page as well.) [[User:Oscarlevin|Oscarlevin]] 22:03, 13 October 2010 (EDT)<br />
<br />
:::: Not a lot of relevant thoughts, but as you probably guessed, that was largely my reason for asking - to suggest things that would be useful to mention on the help pages. [[User:Zerrakhi|Zerrakhi]] 09:33, 17 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=True_or_false&diff=527True or false2010-10-14T00:57:49Z<p>Zerrakhi: </p>
<hr />
<div>Assign a truth value to each of the following statements.<br />
# Statement 5 is false.<br />
# Statement 1 is false.<br />
# Statement 4 is true.<br />
# Only one of these statements is false.<br />
# Statements 2 and 3 are both true or both false.<br />
<br />
<br />
{{needs_hint}}<br />
{{needs_answer}}<br />
{{solution | <br />
<br />
First identify statements with equal vs opposite truth values:<br />
<br />
* 1 has opposite truth value to 5 (from statement 1)<br />
* 2 has opposite truth value to 1 (from statement 2), hence equal truth value to 5 <br />
* 3 has equal truth value to 4 (from statement 3)<br />
* 5 true only if 2 has equal truth value to 3 (from statement 5)<br />
<br />
If 4 false, then:<br />
*3 false (equal value to 4)<br />
*2 has opposite value to 5 (5 true only if 2 false, given 3 false)<br />
But 2 has equal truth value to 5.<br />
So 4 true.<br />
<br />
If 4 true, then:<br />
*3 true (equal value to 4)<br />
*2 and 5 true (otherwise they'd both be false, contradicting 4)<br />
*1 false (opposite value from 2, 5)<br />
<br />
Hence 1 is false and all other statements are true.<br />
<br />
}}<br />
<br />
[[Category: Liar puzzles]]<br />
[[Category: Logic]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Blank_dice&diff=526Blank dice2010-10-13T13:04:09Z<p>Zerrakhi: Hyphenation, internationalise by avoiding brand name, stylistic improvements.</p>
<hr />
<div>Here is a probability brain teaser that I remember from a math for elementary ed class I tutored. <br />
<br />
==Puzzle==<br />
<br />
When you roll two six-sided fair dice, the total number of dots displayed can be anything from 2 to 12. However, some sums are more likely than others. Now suppose you have two six-sided fair dice, with no dots on any of the faces (they are blank). Armed with only a marker pen and your puzzle-solving prowess, how can you draw dots on the dice so that each possible sum will appear with equal probability?<br />
<br />
[[Category: Probability]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Light_bulb&diff=525Light bulb2010-10-13T12:52:41Z<p>Zerrakhi: </p>
<hr />
<div>Here is a classic lateral thinking puzzle.<br />
<br />
==Puzzle==<br />
<br />
A very strange house has a single light bulb connected in the basement. The switch for this light is located on the top floor. Actually, there are three switches on the top floor: two of them do nothing, and the third controls the light in the basement. It is impossible to tell whether the light is on or off when you stand by the switches. How can you determine which switch controls the light bulb while only walking down to the basement one time?<br />
<br />
{{hint | Think about real lightbulbs, not mathematical abstractions.}}<br />
{{needs_answer}}<br />
<br />
[[Category: Lateral thinking]]<br />
[[Category: Cases]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Friends_at_a_party&diff=524Friends at a party2010-10-13T12:46:45Z<p>Zerrakhi: Hint and solution.</p>
<hr />
<div>==Puzzle==<br />
<br />
An extremely popular mathematician threw a party. Including himself, there were 15 people at the party. Must there have been at least two people with the same number of friends present? Explain.<br />
<br />
{{Hint | Assume friendship is mutual, i.e. if A is a friend of B then B is a friend of A.}}<br />
<br />
{{Solution |<br />
<br />
The least number of friends a person might have is 0 (gatecrasher) and the greatest is 14 (friends with everyone, excluding themselves). There are as many integers in that range as there are people at the party, so if everyone has a different number of friends, then every number from 0 to 14 must be equal to that number for some person.<br />
<br />
But if some person (very likely the mathematician) has 14 friends, and we assume friendship is always mutual, then there can be no person with 0 friends. Everyone must have at least one friend, namely the person with 14 friends.<br />
<br />
Therefore the assumption that everyone has a different number of friends leads to a contradiction, and by ''reducto ad absurdum'' we conclude that at least two people must have the same number of friends.<br />
<br />
}}<br />
<br />
[[Category: Graph theory]]<br />
[[Category: Pigeonhole principle]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Friends_at_a_party&diff=523Friends at a party2010-10-13T12:17:31Z<p>Zerrakhi: Clarify that we're only talking about friends who are present at the party!</p>
<hr />
<div>==Puzzle==<br />
<br />
An extremely popular mathematician threw a party. Including himself, there were 15 people at the party. Must there have been at least two people with the same number of friends present? Explain.<br />
<br />
[[Category: Graph theory]]<br />
[[Category: Pigeonhole principle]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Ball_drop&diff=522Ball drop2010-10-13T11:59:36Z<p>Zerrakhi: Spelling and grammar fixes. Also fix link.</p>
<hr />
<div>Here is a nice optimization puzzle from the [http://www.mathsisfun.com: Maths is Fun] website.<br />
<br />
==Puzzle==<br />
<br />
Pool and Billiard mega store Balls-R-Us wants to shoot a commercial for their new Nearly-Indestructible-Billiard-Balls-Are-Amazing brand billiard balls. The commercial will feature a crazed pool shark smashing billiard balls by dropping them from the top of a tall building, only to find that when he drops the NIBBAA balls, they don't break! To make the commercial as convincing as possible, the company wants to use as tall a building as possible, so they need to know the highest floor their billiard balls can be dropped from without breaking.<br />
<br />
While you have a perfectly good 100 storey building to test out the procedure, the producers of the commercial have only given you two billiard balls and want an answer as soon as possible. You realize you could test out each floor in order (first floor, then second floor, and so on), since you can reuse a ball that does not break. But that might take FOREVER! Is there a faster way? What is the least number of times you would have to drop the billiard balls to guarantee finding the highest safe floor?<br />
<br />
==Variations==<br />
<br />
Of course we could ask the same question using a different number of floors, or a different number of balls. In general, what is the least number of test drops needed to guarantee finding the highest safe floor when you have <math>n</math> balls and <math>m</math> total floors.<br />
<br />
==Links==<br />
<br />
[http://www.mathsisfun.com/puzzles/dropping-balls.html Dropping Balls] As described on Maths is Fun.<br />
<br />
[[Category: Optimization puzzles]]<br />
[[Category: Algorithms]]</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Talk:Five_pirates&diff=521Talk:Five pirates2010-10-13T10:13:43Z<p>Zerrakhi: </p>
<hr />
<div>== Error or variation? ==<br />
<br />
Based on comparisons with other versions that I can find online, I believe "''If more than half of the pirates accept the proposal''" should read "''If at least half of the pirates accept the proposal''". <br />
<br />
This is a serious issue, as the puzzle depends critically upon what happens when exactly 50% of pirates accept a proposal. Is it an error, or a variation?</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Talk:Five_pirates&diff=520Talk:Five pirates2010-10-13T10:13:03Z<p>Zerrakhi: Created page with 'Based on comparisons with other versions that I can find online, I believe "''If more than half of the pirates accept the proposal''" should read "''If at least half of the pirat...'</p>
<hr />
<div>Based on comparisons with other versions that I can find online, I believe "''If more than half of the pirates accept the proposal''" should read "''If at least half of the pirates accept the proposal''". <br />
<br />
This is a serious issue, as the puzzle depends critically upon what happens when exactly 50% of pirates accept a proposal. Is it an error, or a variation?</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Five_pirates&diff=519Five pirates2010-10-13T10:12:48Z<p>Zerrakhi: </p>
<hr />
<div>Here is a considerably harder version of the [[Forty thieves]] puzzle. <br />
<br />
==The puzzle==<br />
Five bloodthirsty pirates have just procured 100 gold pieces, which they must now divide up amongst themselves. They agree that the captain will propose a settlement, deciding how many gold pieces each pirate receives. Then all five pirates will vote on the proposal. If more than half of the pirates accept the proposal, it will stand, otherwise the captain must walk the plank. Then, the next highest ranking pirate would propose a new settlement, with the same rules. Each pirate is perfectly logical, greedy and bloodthirsty. How should the captain divide the loot to maximize his take and save his neck?<br />
<br />
{{needs_hint}} {{needs_answer}} {{needs_solution}}</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=518User talk:Oscarlevin2010-10-13T10:12:28Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)<br />
<br />
: The only thing special about puzzles which have "needs ___" on them is that I happened to include that when I first put them up. Some puzzles were supplied before I had created those templates, and others after but I forgot to include the reminder. So for any puzzle you see on here that you think deserves a hint, answer or solution, feel free to provide one.<br />
<br />
: There are other things you can do, if you like. If you know of another puzzle, please add it. You can also improve current puzzles by giving variations, more references, links to similar puzzles, or notes on the mathematical content. In addition to being a collection of puzzles, I hope this wiki becomes a resource for math teachers. So any comments about how a particular puzzle relates to a mathematical concept are welcome. Finally, if you have any further ideas on how to improve the wiki, I'd be happy to hear them. Feedback is always good. [[User:Oscarlevin|Oscarlevin]] 10:18, 12 October 2010 (EDT)<br />
<br />
::I think it would be a good idea to add a page specifically on this topic: a list of ways in which people can contribute. That would also be a good place to summarise the most important templates.<br />
<br />
::Another thing that seems strange is the requirement to log in before viewing a solution. I would question the sense of that, because the audience of people interested in contributing to the wiki is distinct from the audience of people interested in using it as a resource, and members of the latter probably won't bother to create accounts (especially when most puzzles are searchable). But perhaps I misunderstand how you envision the site being used. [[User:Zerrakhi|Zerrakhi]] 06:12, 13 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=516User talk:Oscarlevin2010-10-12T13:38:49Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the [[7 orbs]] puzzle, which I tried to contribute:<br />
<br />
<pre><br />
Let A = set of orbs for which there are 3 or more others the same colour.<br />
Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
Let X = set of orbs for which there are no others the same colour.<br />
<br />
Goal is to prove that some orb is in A.<br />
<br />
Possible combinations:<br />
(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
A A A A A A A<br />
A A A A A A X<br />
A A A A A B B<br />
A A A A A X X<br />
A A A A B B B<br />
A A A A B B X<br />
A A A A X X X<br />
<br />
Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
If two tested pairs glow the same colour, all members of those pairs are in A<br />
If only one of the three tested pairs glows, members of that pair are in A<br />
If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
If no tested pairs glow, orb 7 is in A.<br />
<br />
In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</pre><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)<br />
<br />
== More questions ==<br />
<br />
I'm wondering why only a few puzzles are in the [[:Category:Needs_solution | needs solution]] category, and likewise for other similar categories. What's special about ''those'' puzzles in particular? <br />
<br />
Because the [[7 orbs]] puzzle is listed first on the front page, I assumed it was representative, and that all puzzles would have a hint, answer and solution in the same format. Now that I've seen that's not the case, I'm less sure about how I can contribute further. (Except by asking pointed questions, of course.) [[User:Zerrakhi|Zerrakhi]] 09:38, 12 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=490User talk:Oscarlevin2010-10-11T23:27:24Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the "7_orbs" puzzle, which I tried to contribute:<br />
<br />
<small><br />
:Let A = set of orbs for which there are 3 or more others the same colour.<br />
:Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
:Let X = set of orbs for which there are no others the same colour.<br />
<br />
:Goal is to prove that some orb is in A.<br />
<br />
:Possible combinations:<br />
:(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
:A A A A A A A<br />
:A A A A A A X<br />
:A A A A A B B<br />
:A A A A A X X<br />
:A A A A B B B<br />
:A A A A B B X<br />
:A A A A X X X<br />
<br />
:Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
:Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
:If two tested pairs glow the same colour, all members of those pairs are in A<br />
:If only one of the three tested pairs glows, members of that pair are in A<br />
:If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
:If no tested pairs glow, orb 7 is in A.<br />
<br />
:In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
:The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</small><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
: Yes that is the sort of thing I'm looking for. I'll see if I can fix the multiple lines problem. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
: So I can't seem to replicate your problem. Is it just that the lines are not breaking? Like this:<br />
{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}<br />
: Because that can be fixed will a line break between them, as in:<br />
<br />
{{Solution | <br />
solution<br />
<br />
over<br />
<br />
multiple<br />
<br />
lines<br />
}}<br />
: Or is there some other problem? [[User:Oscarlevin|Oscarlevin]] 15:25, 11 October 2010 (EDT)<br />
::The problem wasn't with your multiple lines, it was with the = sign. Templates allow you to have named parameters, which you assign with =. I don't know if there is a workaround for this. [[User:Oscarlevin|Oscarlevin]] 15:41, 11 October 2010 (EDT)<br />
<br />
::: Ah. Never thought of that. I just tested in preview mode, and found that nowiki tags around the = sign seemed to work. I've written something up on the [[Help:Editing | Editing Help]] page. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)<br />
<br />
: Did the email you get contain a bunch of <nowiki><math></math></nowiki> tags? I'm using an extension that switches dollar signs to math mode (ala LaTeX), and it has screwed up a lot of the system messages. I'll see if I can fix this too. [[User:Oscarlevin|Oscarlevin]] 15:06, 11 October 2010 (EDT)<br />
<br />
:: Yes, that's right. [[User:Zerrakhi|Zerrakhi]] 19:27, 11 October 2010 (EDT)</div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=Help:Editing&diff=489Help:Editing2010-10-11T23:26:29Z<p>Zerrakhi: Some helpful assistance</p>
<hr />
<div>== General assistance ==<br />
<br />
Pending the completion of this page, see the [http://en.wikipedia.org/wiki/Wikipedia:Cheatsheet editing help page on Wikipedia] for further editing assistance, and the [http://en.wikipedia.org/wiki/Help:Wiki_markup Wiki markup page] for even more information.<br />
<br />
On talk pages, sign comments with <nowiki>~~~~</nowiki>.<br />
<br />
== Templates ==<br />
<br />
=== Use of the equals sign ===<br />
<br />
Within the body of a template, such as the '''<nowiki>{{Solution | ... }}</nowiki>''' template, the wiki will interpret an '''=''' sign as indicating a named parameter, which will lead to the template not being displayed correctly. Therefore, either avoid the use of the '''=''' sign, or surround it in nowiki tags, like this:<br />
<br />
<pre>{{Solution |<br />
1+1<nowiki><</nowiki>nowiki<nowiki>></nowiki>=<nowiki></</nowiki>nowiki<nowiki>></nowiki>2<br />
}}</pre></div>Zerrakhihttps://mathpuzzlewiki.com/index.php?title=User_talk:Oscarlevin&diff=481User talk:Oscarlevin2010-10-11T16:27:29Z<p>Zerrakhi: </p>
<hr />
<div>== Solutions over multiple lines ==<br />
<br />
I visited the site following the link in your comments on Richard Wiseman's most recent Friday Puzzle solution, and thought I'd see if I could contribute.<br />
<br />
However, I can't contribute a solution because the wiki won't accept a solution that takes more than one line. For example, this doesn't work:<br />
<br />
<pre>{{Solution | <br />
solution<br />
over<br />
multiple<br />
lines<br />
}}</pre><br />
<br />
Pretty much all solutions will require multiple lines, so this is a real problem. I assume a "solution" means an answer with working, whereas an "answer" is just the answer by itself.<br />
<br />
Below is my solution to the "7_orbs" puzzle, which I tried to contribute:<br />
<br />
<small><br />
:Let A = set of orbs for which there are 3 or more others the same colour.<br />
:Let B = set of orbs for which there are 1 or 2 others the same colour.<br />
:Let X = set of orbs for which there are no others the same colour.<br />
<br />
:Goal is to prove that some orb is in A.<br />
<br />
:Possible combinations:<br />
:(using the shorthand that A denotes an orb that's in A, etc)<br />
<br />
:A A A A A A A<br />
:A A A A A A X<br />
:A A A A A B B<br />
:A A A A A X X<br />
:A A A A B B B<br />
:A A A A B B X<br />
:A A A A X X X<br />
<br />
:Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.<br />
<br />
:Comparing the results of testing those three pairs with the possible combinations, you can easily show that:<br />
<br />
:If two tested pairs glow the same colour, all members of those pairs are in A<br />
:If only one of the three tested pairs glows, members of that pair are in A<br />
:If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A<br />
:If no tested pairs glow, orb 7 is in A.<br />
<br />
:In all cases, three tests suffice to prove that some orb is in A.<br />
<br />
:The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.<br />
</small><br />
<br />
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)<br />
<br />
== No link in email ==<br />
<br />
The email received upon registration, the one containing the words "open this link in your browser", does not actually contain any links or URLs to open! [[User:Zerrakhi|Zerrakhi]] 12:27, 11 October 2010 (EDT)</div>Zerrakhi