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.
Below is my solution to the
"7_orbs" puzzle, which I tried to contribute:
small> :Let A = set of orbs for which there are 3 or more others the same colour. :Let B = set of orbs for which there are 1 or 2 others the same colour. :Let X = set of orbs for which there are no others the same colour.
:Goal is to prove that some orb is in A.
:Possible combinations: :(using the shorthand that A denotes an orb that's in A, etc)
:A A A A A A A :A A A A A A X :A A A A A B B :A A A A A X X :A A A A B B B :A A A A B B X :A A A A X X X
:Label the orbs 1, 2, 3, 4, 5, 6, 7 and test 1&2, 3&4, 5&6.
:Comparing the results of testing those three pairs with the possible combinations, you can easily show that:
:If two tested pairs glow the same colour, all members of those pairs are in A :If only one of the three tested pairs glows, members of that pair are in A :If two tested pairs glow different colours and the other doesn't glow, orb 7 is in A :If no tested pairs glow, orb 7 is in A.
:In all cases, three tests suffice to prove that some orb is in A.
:The only other thing required is to prove that two tests are not necessarily sufficient, which is trivial.</ small>
Is that the sort of thing you're looking for? [[User:Zerrakhi|Zerrakhi]] 12:16, 11 October 2010 (EDT)
::: This should be fixed now. Just testing a few things out. [[User:Test1|Test1]] 22:03, 11 October 2010 (EDT)