https://mathpuzzlewiki.com/api.php?action=feedcontributions&user=Oscarlevin&feedformat=atomMath Puzzle Wiki - User contributions [en]2024-03-29T11:13:23ZUser contributionsMediaWiki 1.29.0https://mathpuzzlewiki.com/index.php?title=Flying_trains&diff=1415Flying trains2023-10-05T14:53:03Z<p>Oscarlevin: /* Puzzle */</p>
<hr />
<div>[[File:Train template.svg|right|150px]]<br />
<br />
Here is the classic not-really-a-calculus puzzle.<br />
<br />
==Puzzle==<br />
<br />
Towns A and B are connected by a single railroad track, exactly 210 miles long. One fateful day, at exactly 1:00pm, a red train leaves town A traveling to town B at 40 miles per hour. At the same time, a bright blue train leaves town B traveling to town A at 30 miles per hour. As the red train starts to move, a brave fly takes off of the windshield and flies at 55 miles per hour towards town B. As soon as the fly reaches the blue train, he immediately changes direction and flies back towards town A, again, traveling at 55 miles per hour. When he gets to the red train, he changes direction again. The fly continues to fly back and forth between the two, ever nearing trains until he is smashed to bits when the trains sadly collide.<br />
<br />
How far did the fly fly between 1:00pm and his all-to-early death?<br />
<br />
==Help==<br />
<br />
{{Hint | You certainly could try to set up some sort of infinite sum, but there is an easier way. First answer this: how ''long'' did the fly travel?}}<br />
{{Answer | 165 miles.}}<br />
{{Solution | Given the velocities of the trains, the distance between them is decreasing at a rate of 70 miles per hour. Thus the trains will collide in exactly 3 hours. The fly will be traveling at 55 miles per hour for this entire 3 hour period, which comes to 165 miles traveled.}}<br />
<br />
==See also==<br />
<br />
[[Girl, boy and dog]] <br />
<br />
[[Category: Velocity puzzles]]<br />
[[Category: Calculus]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Ball_drop&diff=1391Ball drop2019-09-03T21:33:48Z<p>Oscarlevin: /* Puzzle */</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 story 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 <m>n</m> balls and <m>m</m> 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>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Meta:Links&diff=1390Meta:Links2019-07-18T04:00:37Z<p>Oscarlevin: </p>
<hr />
<div>Looking for more puzzles? Here are some problem solving related resources.<br />
<br />
==Collections and Blogs==<br />
<br />
*[https://www.theguardian.com/science/series/alex-bellos-monday-puzzle Monday Puzzle Blog] by Alex Bellos.<br />
*[https://www.popularmechanics.com/riddles-logic-puzzles/ Riddle of the Week] from Popular Mechanics.<br />
*[https://blogs.wsj.com/puzzle/category/varsity-math-2/ Varsity Math] Puzzle block of the Wall Street Journal.<br />
*[https://projecteuler.net/ Project Euler]. A series of challenging math and computer programming problems.<br />
*[http://www.mathpuzzle.com/ MathPuzzle.com]<br />
*[https://blog.tanyakhovanova.com/category/puzzles/ Tanya Khovanova's Math Blog Puzzles].<br />
*{{Car Talk}}<br />
*{{Mathisfun}}<br />
*{{Mathcentral}}<br />
*{{pzzls}}<br />
*[https://www.unco.edu/nhs/mathematical-sciences/challenge/ The Math Challenge Problem] from the University of Norther Colorado.<br />
<br />
==Videos==<br />
<br />
*[https://www.youtube.com/playlist?list=PLJicmE8fK0EhMjOWNNhlY4Lxg8tupXKhC TED-Ed Riddles on YouTube]. Nice animations for classic puzzles.<br />
*[https://www.youtube.com/user/MindYourDecisions/featured Mind Your Decisions]. See also the corresponding [https://mindyourdecisions.com/blog/ blog].<br />
*[https://www.youtube.com/user/numberphile/featured Numberphile]. Lots of interesting math, including a number of good puzzles.<br />
<br />
<br />
==Podcasts==<br />
<br />
*{{Math Factor}} A math puzzle podcast from a while back.<br />
*[https://www.npr.org/series/4473090/sunday-puzzle The Sunday Puzzle on NPR]. Mostly word puzzles, but occasionally an interesting math nugget.<br />
<br />
==Books==<br />
<br />
*{{Problem Solving}}<br />
*{{Averbach}}<br />
*{{Martin Gardner books}}<br />
*{{Winkler book}}. <br />
*{{Smullyan riddle}} Very nice collection of puzzles, all tied together in a story. Plus, some logic content as well.<br />
*{{Sideways Arithmetic}} A collection of cryptarithmetic puzzles.<br />
<br />
==Games==<br />
<br />
*{{Mindtrap}}<br />
*{{Professor Layton}}<br />
*{{Professor Layton 2}}<br />
<br />
==Problem Solving and Teaching Resources==<br />
<br />
*[https://www.mathteacherscircle.org/resources/math-sessions/ Math Teacher Circles Sessions].<br />
<br />
__NOTOC__</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Meta:Links&diff=1389Meta:Links2019-07-18T02:52:26Z<p>Oscarlevin: </p>
<hr />
<div>Looking for more puzzles? Here are some problem solving related resources.<br />
<br />
==Collections and Blogs==<br />
<br />
*[https://www.theguardian.com/science/series/alex-bellos-monday-puzzle Monday Puzzle Blog] by Alex Bellos.<br />
*[http://www.mathpuzzle.com/ MathPuzzle.com]<br />
*{{Car Talk}}<br />
*{{Mathisfun}}<br />
*{{Mathcentral}}<br />
*{{pzzls}}<br />
*[https://www.unco.edu/nhs/mathematical-sciences/challenge/ The Math Challenge Problem] from the University of Norther Colorado.<br />
<br />
==Videos==<br />
<br />
*[https://www.youtube.com/playlist?list=PLJicmE8fK0EhMjOWNNhlY4Lxg8tupXKhC TED-Ed Riddles on YouTube]. Nice animations for classic puzzles.<br />
*[https://www.youtube.com/user/MindYourDecisions/featured Mind Your Decisions]. See also the corresponding [https://mindyourdecisions.com/blog/ blog].<br />
*[https://www.youtube.com/user/numberphile/featured Numberphile]. Lots of interesting math, including a number of good puzzles.<br />
<br />
<br />
==Podcasts==<br />
<br />
*{{Math Factor}} A math puzzle podcast from a while back.<br />
*[https://www.npr.org/series/4473090/sunday-puzzle The Sunday Puzzle on NPR]. Mostly word puzzles, but occasionally an interesting math nugget.<br />
<br />
==Books==<br />
<br />
*{{Problem Solving}}<br />
*{{Averbach}}<br />
*{{Martin Gardner books}}<br />
*{{Winkler book}}. <br />
*{{Smullyan riddle}} Very nice collection of puzzles, all tied together in a story. Plus, some logic content as well.<br />
*{{Sideways Arithmetic}} A collection of cryptarithmetic puzzles.<br />
<br />
==Games==<br />
<br />
*{{Mindtrap}}<br />
*{{Professor Layton}}<br />
*{{Professor Layton 2}}<br />
<br />
==Problem Solving and Teaching Resources==<br />
<br />
*[https://www.mathteacherscircle.org/resources/math-sessions/ Math Teacher Circles Sessions].<br />
<br />
__NOTOC__</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Template:Bradley_PoW&diff=1388Template:Bradley PoW2019-07-18T02:22:41Z<p>Oscarlevin: </p>
<hr />
<div>Bradley University's Problem of the Week (no longer available).</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Meta:Links&diff=1387Meta:Links2019-07-18T02:21:23Z<p>Oscarlevin: </p>
<hr />
<div>Looking for more puzzles? Here are some problem solving related resources.<br />
<br />
<br />
==Websites==<br />
<br />
*{{Math Factor}} A math puzzle podcast.<br />
*{{Car Talk}}<br />
*{{Mathisfun}}<br />
*{{Mathcentral}}<br />
*{{pzzls}}<br />
*{{Richard Wiseman}}<br />
*{{Bradley PoW}} No longer active, but there are 10 years worth of archives. From professor Alberto L. Delgado.<br />
<br />
==Books==<br />
<br />
*{{Problem Solving}}<br />
*{{Averbach}}<br />
*{{Martin Gardner books}}<br />
*{{Winkler book}}. <br />
*{{Smullyan riddle}} Very nice collection of puzzles, all tied together in a story. Plus, some logic content as well.<br />
*{{Sideways Arithmetic}} A collection of cryptarithmetic puzzles.<br />
<br />
==Games==<br />
<br />
*{{Mindtrap}}<br />
*{{Professor Layton}}<br />
*{{Professor Layton 2}}</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Ants_on_a_cube&diff=1383Ants on a cube2017-10-26T15:15:32Z<p>Oscarlevin: Created page with "This is a generalization of the first puzzle in Presh Talwalkar's ''Math Puzzles Volume 1''. == Puzzle == 8 ants sit on the vertices of a cube (floating in space). Each ant..."</p>
<hr />
<div>This is a generalization of the first puzzle in Presh Talwalkar's ''Math Puzzles Volume 1''.<br />
<br />
== Puzzle ==<br />
<br />
8 ants sit on the vertices of a cube (floating in space). Each ant sets off along one of the edges incident to its vertex, at random. What is the probability that no ants will collide?</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Cents_per_cents&diff=1382Cents per cents2017-10-01T19:15:04Z<p>Oscarlevin: </p>
<hr />
<div>This is my version of a puzzle I originally saw on [https://www.theguardian.com/science/2017/jun/19/can-you-solve-it-pythagorass-best-puzzles Alex Bellos's Puzzle Blog].<br />
<br />
==Puzzle==<br />
<br />
100 of the dimmest bank robbers recently broke in to the Denver Mint and made off with a truck-load of fresh new pennies. To divvy up their loot, they decide that the youngest thief will get 1% of the score, then the 2nd youngest will get 2% of what is left, the third youngest getting 3% of what is left after that, and so on, until the oldest (100th youngest) gets 100% of what is left after everyone has taken their share. <br />
<br />
Which thief receives the largest share? Further, what is the least amount of money he will leave with, assuming no rounding occurs while splitting the take?</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Cents_per_cents&diff=1381Cents per cents2017-08-06T14:32:28Z<p>Oscarlevin: Created page with "This is my version of a puzzle I originally saw on [https://www.theguardian.com/science/2017/jun/19/can-you-solve-it-pythagorass-best-puzzles Alex Bellos's Puzzle Block]. ==P..."</p>
<hr />
<div>This is my version of a puzzle I originally saw on [https://www.theguardian.com/science/2017/jun/19/can-you-solve-it-pythagorass-best-puzzles Alex Bellos's Puzzle Block].<br />
<br />
==Puzzle==<br />
<br />
100 of the dimmest bank robbers recently broke in to the Denver Mint and made off with a truck-load of fresh new pennies. To divvy up their loot, they decide that the youngest thief will get 1% of the score, then the 2nd youngest will get 2% of what is left, the third youngest getting 3% of what is left after that, and so on, until the oldest (100th youngest) gets 100% of what is left after everyone has taken their share. <br />
<br />
Which thief receives the largest share? Further, what is the least amount of money he will leave with, assuming no rounding occurs while splitting the take?</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Next_letter&diff=1380Next letter2017-07-25T13:16:29Z<p>Oscarlevin: added links.</p>
<hr />
<div>A nice short puzzle, perfect for posing at the start of class.<br />
<br />
==Puzzle==<br />
<br />
What are the next letters in the sequence: O T T F F S S E?<br />
<br />
==Help==<br />
<br />
{{Hint|The next letter is N}}<br />
<br />
{{Answer|N T E T T ...}}<br />
<br />
{{Solution|The letters in the sequence are the initials of the natural numbers: One, Two, Three, etc. So the sequence continues: N T E T T F F S S E N T.}}<br />
<br />
==See Also==<br />
<br />
[[Next sequence]]<br />
<br />
[[Sequence next in sequence]]<br />
<br />
[[Category: Lateral thinking]]<br />
[[Category: Short puzzles]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Next_sequence&diff=1379Next sequence2017-07-25T13:14:21Z<p>Oscarlevin: </p>
<hr />
<div>Here is a classic sequence guessing puzzle.<br />
<br />
==Puzzle==<br />
<br />
Consider the sequence 1, 11, 21, 1211, 111221, 312211... What comes next?<br />
<br />
==Help==<br />
<br />
{{Answer| 13112221.}}<br />
<br />
==See Also==<br />
<br />
[[Sequence next in sequence]]<br />
<br />
<br />
[[Category: Lateral thinking]]<br />
[[Category: Sequences]]<br />
[[Category: Short puzzles]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Sequence_next_in_sequence&diff=1378Sequence next in sequence2017-07-25T13:10:46Z<p>Oscarlevin: Created page with "This sequence guessing puzzle hides a beautiful, well known sequence. ==Puzzle== What comes next: 1, 2, 2, 1, 1, 2, 1, ....? ==Help== {{Hint| Count the number of repeating..."</p>
<hr />
<div>This sequence guessing puzzle hides a beautiful, well known sequence.<br />
<br />
==Puzzle==<br />
<br />
What comes next: 1, 2, 2, 1, 1, 2, 1, ....?<br />
<br />
==Help==<br />
<br />
{{Hint| Count the number of repeating digits.}}<br />
<br />
{{Answer| 2}}<br />
<br />
{{Solution| This is the famous Kolakoski sequence (see the wikipedia page listed below). The sequence self-encodes the run length of repeated digits. These run lengths are 1, 2, 2, 1, 1, 2, 1,...., which is the sequence again. }}<br />
<br />
==Bonus==<br />
<br />
Find a sequence using the numbers 1, 2, and 3 that has the same property as the one above.<br />
<br />
==References==<br />
<br />
[https://en.wikipedia.org/wiki/Kolakoski_sequence Wikipedia page] (contains spoilers).<br />
<br />
[[Category:Short puzzles]]<br />
[[Category:Sequences]]<br />
<br />
__NOTOC__</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1376Wagon collision2017-07-25T02:19:31Z<p>Oscarlevin: /* Help */</p>
<hr />
<div>This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov. <br />
<br />
==Puzzle==<br />
<br />
Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?<br />
<br />
==Help==<br />
<br />
{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) : 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of \(I\) to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x{{=}}1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than \(2R\) and each wagon has radius exactly \(R\) both wagons must collide at this point.<br />
}}<br />
<br />
==References==<br />
<br />
[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1375Wagon collision2017-07-25T02:18:17Z<p>Oscarlevin: /* Help */</p>
<hr />
<div>This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov. <br />
<br />
==Puzzle==<br />
<br />
Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?<br />
<br />
==Help==<br />
<br />
{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) \suchthat 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x{{=}}1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.<br />
}}<br />
<br />
==References==<br />
<br />
[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1374Wagon collision2017-07-25T02:17:41Z<p>Oscarlevin: /* Help */</p>
<hr />
<div>This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov. <br />
<br />
==Puzzle==<br />
<br />
Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?<br />
<br />
==Help==<br />
<br />
{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) \suchthat 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.<br />
}}<br />
<br />
==References==<br />
<br />
[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1373Wagon collision2017-07-25T02:17:19Z<p>Oscarlevin: /* Help */</p>
<hr />
<div>This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov. <br />
<br />
==Puzzle==<br />
<br />
Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?<br />
<br />
==Help==<br />
<br />
{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) {{|}} 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.<br />
}}<br />
<br />
==References==<br />
<br />
[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1372Wagon collision2017-07-25T02:16:53Z<p>Oscarlevin: /* Help */</p>
<hr />
<div>This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov. <br />
<br />
==Puzzle==<br />
<br />
Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?<br />
<br />
==Help==<br />
<br />
{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) : 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.<br />
}}<br />
<br />
==References==<br />
<br />
[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1371Wagon collision2017-07-25T02:16:38Z<p>Oscarlevin: /* Help */</p>
<hr />
<div>This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov. <br />
<br />
==Puzzle==<br />
<br />
Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?<br />
<br />
==Help==<br />
<br />
{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y) \| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.<br />
}}<br />
<br />
==References==<br />
<br />
[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1370Wagon collision2017-07-25T02:16:13Z<p>Oscarlevin: /* Help */</p>
<hr />
<div>This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov. <br />
<br />
==Puzzle==<br />
<br />
Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?<br />
<br />
==Help==<br />
<br />
{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I {{=}} \{(x,y)| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y {{=}} 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.<br />
}}<br />
<br />
==References==<br />
<br />
[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Wagon_collision&diff=1369Wagon collision2017-07-25T02:15:45Z<p>Oscarlevin: /* Help */</p>
<hr />
<div>This puzzle appears in V. I. Arnold's classical text on ODEs, where it's attributed to N.N. Konstantinov. <br />
<br />
==Puzzle==<br />
<br />
Suppose there are cities A and B connected to each other by two non-intersecting roads. Furthermore, suppose we know that two cars attached by a rope of length less than 2R are able to travel together on different roads from City A to City B without the rope tearing. Given this, is it possible for two circular wagons, each of radius R, each traveling along its center and each starting in a different city, to travel in opposite directions along different roads without colliding as they pass?<br />
<br />
==Help==<br />
<br />
{{Solution| Let <m>x</m> denote the distance between City A and a vehicle (so either a wagon or a car) traveling on one of the roads. Similarly, let <m>y</m> denote the distance between City A and the other vehicle traveling on the other road. We can imagine these values together representing points in the unit square <m>I = \{(x,y)| 0 \leq x,y \leq 1\}</m>. In the case of the cars, both vehicles start at City A and so <m>x{{=}}y{{=}}0</m> initially. As both cars traverse their respective roads to City B a continuous curve is drawn out linking the bottom left corner of $I$ to the top right corner. In the case of the wagons, both vehicles start at different cities, say <m>x=1</m> and <m>y = 0</m>. Thus, as both wagons move in opposite directions <m>x</m> is tending towards 0 (to City A) and <m>y</m> is tending towards 1 (to City B), and so a continuous curve is traced out from the bottom right corner to the top left corner. Since both curves link different corners they must intersect at some point in $I$. At this point of intersection each wagon must be at the exact position each car was at on their respective roads. Since the length of the rope attaching both cars was less than 2R and each wagon has radius exactly R both wagons must collide at this point.<br />
}}<br />
<br />
==References==<br />
<br />
[http://www.amazon.com/Ordinary-Differential-Equations-V-Arnold/dp/0262510189 Ordinary Differential Equations] - V. I. Arnold's book on ODEs.</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=One-way_roads&diff=1368One-way roads2017-07-25T02:11:30Z<p>Oscarlevin: /* Puzzle */</p>
<hr />
<div>Found this one on the SUNY Stony Brook Math Problem of the Month archives.<br />
<br />
==Puzzle==<br />
<br />
A county has <m>n > 4</m> cities. Is it possible to connect some pairs of cities by one-way roads so that one can travel from every city to every other city using only one or two roads? If so, how? Note that for every pair of cities A and B, only one road connecting A with B is allowed; this road leads from A to B or from B to A, but not both ways.<br />
<br />
[[Category: Graph theory]]<br />
[[Category: New]]<br />
[[Category: Needs solution]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=One-way_roads&diff=1367One-way roads2017-07-25T02:10:58Z<p>Oscarlevin: /* Puzzle */</p>
<hr />
<div>Found this one on the SUNY Stony Brook Math Problem of the Month archives.<br />
<br />
==Puzzle==<br />
<br />
A county has <math>n > 4</math> cities. Is it possible to connect some pairs of cities by one-way roads so that one can travel from every city to every other city using only one or two roads? If so, how? Note that for every pair of cities A and B, only one road connecting A with B is allowed; this road leads from A to B or from B to A, but not both ways.<br />
<br />
[[Category: Graph theory]]<br />
[[Category: New]]<br />
[[Category: Needs solution]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Condominium&diff=1364Condominium2015-04-05T01:15:19Z<p>Oscarlevin: </p>
<hr />
<div>==Puzzle==<br />
<br />
In a certain condominium community, 2/3 of all the women are married (to men) and 3/5 of all the men are married (to women). What fraction of the entire condominium community is married?<br />
<br />
{{Solution| The fraction of married women (M) to unmarried women (U) and the fraction of married men (M) to unmarried men (U) can be visualized as follows.<br />
<br />
MMMMMMUUU (Women)<br />
<br />
MMMMMMUUUU (Men)<br />
<br />
We know that the number of women who are married equals the number of men who are married. So the women’s M region and the men’s M region must be the same size. By using the same units in both the women’s diagram and the men’s diagram, we see that the two M regions are six units long each. This represents a total of 12 units out of a total possible of 19 units (9 from the women plus 10 from the men). So 12/19 of the condominium community is married.}}<br />
<br />
[[Category: Algebra]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=File:Trolls.png&diff=1363File:Trolls.png2015-03-31T16:21:55Z<p>Oscarlevin: </p>
<hr />
<div></div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Too_many_trolls&diff=1362Too many trolls2015-03-31T16:20:42Z<p>Oscarlevin: Created page with "200px This is a logic puzzle I made up for UNC's Math Challenge Problem ==Puzzle== While walking through a fictional forest, you come upon a large..."</p>
<hr />
<div>[[File:trolls.png|right|200px]]<br />
<br />
This is a logic puzzle I made up for UNC's Math Challenge Problem<br />
<br />
==Puzzle==<br />
<br />
While walking through a fictional forest, you come upon a large group of trolls. The trolls all look identical, but you know that some of the trolls are knights who always tell the truth, while the rest of the trolls are knaves who always lie. Each troll makes a single statement. <br />
<br />
The first troll says, "Hi, I'm Tucker."<br />
<br />
The remaining 41 trolls each say, in order, "If the previous troll is a knight, then by the time I'm done speaking you will have heard more lies that truths."<br />
<br />
Is the first troll's name really Tucker? And which of the remaining trolls are knight and which are knaves?<br />
<br />
==Help==<br />
<br />
{{Hint| Suppose the first troll is a knave. What does this tell you about the second troll's statement? Conclude that the third troll is impossible.}}<br />
<br />
{{Answer| Troll 1 is named Tucker. Trolls will then alternate between being knaves and knights.}}<br />
<br />
{{Solution| Suppose that the first troll is a knave. This makes the second troll's statement true, since the hypothesis of his implication is false. Thus troll 2 is a knight. Troll 3 cannot be a knight for if he were, then it would follow that by the time he was done speaking, you would have heard more lies than truths, but you would have heard two truths and one lie. But also the Troll 3 cannot be a knave, for this would make his hypothesis true and conclusion false, meaning that you had not in fact heard more lies than truths, but you had heard two lies and one truth, a contradiction. Thus it is impossible for the first troll to be a knave.<br />
<br />
So the first troll really is a knight (and thus named Tucker). This makes the second troll's statement false (as the previous troll really is a knight and you will not have heard more lies than truths). The third troll will then be telling the truth (the second troll is not a knight). Troll 4 will be lying again, and so on. Every odd-numbered troll will be a knight and every even numbered troll will be a knave.}}<br />
<br />
==See also==<br />
<br />
*[[Chests of logic]]<br />
*[[Chests of logic 4]]<br />
*[[Two guards]]<br />
*[[Three princesses]]<br />
<br />
[[Category: Logic]]<br />
[[Category: Cases]]<br />
[[Category: New]]<br />
[[Category: MCP]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Rabbit_row&diff=1360Rabbit row2014-11-16T18:21:01Z<p>Oscarlevin: Created page with "Here is a puzzle based on a neat graph theory counting problem. ==Puzzle== Eleven white rabbits live in eleven white houses, all in a row. One day, the rabbits get together..."</p>
<hr />
<div>Here is a puzzle based on a neat graph theory counting problem.<br />
<br />
==Puzzle==<br />
<br />
Eleven white rabbits live in eleven white houses, all in a row. One day, the rabbits get together and decide that they should spruce up their rabbit row by painting some or all of their houses. They decide that while they don't necessarily need to paint every single house, they will definitely NOT leave any two adjacent houses white. How many choices do they have for which collection of houses to paint?<br />
<br />
==Help==<br />
<br />
{{Hint | Try solving the pattern for smaller numbers of houses and look for a pattern.}}<br />
<br />
{{Answer | There are 233 different collections of houses which could be painted.}}<br />
<br />
{{Solution | Perhaps surprisingly, if you start with <m>n</m> houses, the number of collections of houses which could be painted given this restriction is the <m>n+2</m>nd Fibonacci number. To see this, note that with 1 house, there are 2 collections (either paint or don't paint the one house). With 2 houses, there are 3 collections (paint the first, second, or both houses). Now inductively suppose that you want to paint <m>n</m> houses. You could either paint or not paint the first house. If you paint the first house, the remaining <m>n-1</m> houses need to be painted, and we know how to do that. If you don't paint the first house, then you ''must'' paint the second house, and then have your choice of how to paint the remaining <m>n-2</m> houses, which we know how to count.}}<br />
<br />
==Variations==<br />
<br />
Another group of eleven white rabbits also live in eleven white houses, but these are positioned in a large circle. Again, they want to repaint some or all of the houses, leaving no two adjacent houses white. How many ways can they do this?<br />
<br />
Of course, we could also ask these questions and include paint color choices. For example, what if every house would be painted red, white or blue, but we don't want any two adjacent houses to be colored identically. How many choices do the rabbits have?<br />
<br />
==Mathematics==<br />
<br />
The original puzzle asks for the number of independent sets of the path graph <m>P_{11}</m>. An independent set is a set of vertices in a graph no two of which are adjacent (connected by an edge). The first variation asks for the number of independent sets in a cycle graph. The second variation asks for the number of proper 3-colorings of such graphs.<br />
<br />
<br />
[[Category: Combinatorics]]<br />
[[Category: Graph theory]]<br />
[[Category: Induction]]<br />
[[Category: Sequences]]<br />
<br />
__NOTOC__</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Domino_circuit&diff=1359Domino circuit2014-11-10T22:46:50Z<p>Oscarlevin: Created page with "==Puzzle== A domino consists of two squares, each with some number of dots between 0 and 6 in each square. A standard ''double-six'' set of dominoes contains exactly one domi..."</p>
<hr />
<div>==Puzzle==<br />
<br />
A domino consists of two squares, each with some number of dots between 0 and 6 in<br />
each square. A standard ''double-six'' set of dominoes contains exactly one domino with each possible<br />
pair of numbers on it. Suppose you start laying down a line of dominoes, observing the<br />
rule that two dominoes can touch only if the numbers on the touching squares are equal. After laying down all but the last domino you notice that the two ends happen to have 3 and 5 dots respectively. What does the last domino look like?<br />
<br />
[[Category: MCP]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Penguin_lineups&diff=1358Penguin lineups2014-11-10T22:39:21Z<p>Oscarlevin: Created page with "Here is a counting problem based on Gilbreath permutations. ==Puzzle== You have 10 penguins, each a different height. You want to take a photo of the penguins in a single s..."</p>
<hr />
<div>Here is a counting problem based on Gilbreath permutations.<br />
<br />
==Puzzle==<br />
<br />
You have 10 penguins, each a different height. You want to take a photo of the penguins in a single straight line. First though, you select some number of the penguins to be looking slightly to the right, and the others to be pointing slightly to the left. For fun, you decide that all the right-looking penguins should be increasing in height while all the left-looking penguins should be decreasing in height, as you move from right to left. How many different such arrangements are possible?<br />
<br />
[[Category: Combinatorics]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Five_card_draw&diff=1341Five card draw2014-11-10T03:16:25Z<p>Oscarlevin: Created page with "This puzzle was inspired by a magic trick performed by Ricky Jay on the tonight show. ==Puzzle== Take a well shuffled deck of cards and deal off 10, face up, in a single lin..."</p>
<hr />
<div>This puzzle was inspired by a magic trick performed by Ricky Jay on the tonight show.<br />
<br />
==Puzzle==<br />
<br />
Take a well shuffled deck of cards and deal off 10, face up, in a single line. You and a friend are going to take turns taking a card from either end of the line. The player with the better five card hand wins. Do you want to go first or second?<br />
<br />
{{Hint | With a little planning (and false shuffling) you can make this into more of a magic trick by dealing the cards face down.}}<br />
<br />
==See also==<br />
<br />
*[[Coin game]]<br />
<br />
[[Category: New]]<br />
[[Category: Algorithms]]<br />
[[Category: Combinatorics]]</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=MediaWiki:Sidebar&diff=1340MediaWiki:Sidebar2014-11-09T03:18:39Z<p>Oscarlevin: Undo revision 1337 by Oscarlevin (talk)</p>
<hr />
<div>* Navigation<br />
** mainpage|mainpage-description<br />
** Special:Categories|Categories<br />
** Special:AllPages|All puzzles<br />
** Meta:Links|Links<br />
** randompage-url|Random puzzle<br />
<br />
* SEARCH<br />
<br />
* Contribute<br />
** Help:Contents|Help<br />
** Help:Ways to contribute|Ways to contribute<br />
** Special:RecentChanges|Recent changes</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Help:Contents&diff=1339Help:Contents2014-11-09T03:17:23Z<p>Oscarlevin: </p>
<hr />
<div>These are pages designed to help in the maintenance of the site. General help on using a wiki can be found [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents here]. If you are looking for help on a specific puzzle, you are on you own.<br />
<br />
==Help pages==<br />
<br />
{{Special:Allpages/Help:}}</div>Oscarlevinhttps://mathpuzzlewiki.com/index.php?title=Help:Contents&diff=1338Help:Contents2014-11-09T03:16:42Z<p>Oscarlevin: </p>
<hr />
<div>These are pages designed to help in the maintenance of the site. General help on using a wiki can be found [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents| here]. If you are looking for help on a specific puzzle, you are on you own.<br />
<br />
==Help pages==<br />
<br />
{{Special:Allpages/Help:}}</div>Oscarlevin