[mcphage]

This one is my favorite, too—it really highlights what the job of a mathematician is. The board is the board, and the dominos either fit or they don’t, and it’s not clear why. But once someone adds the checkerboard shading—not changing the problem at all, but just adding a new way to look at it—suddenly the solution falls out, clear and obviously true.

[brennanpeterson]

I am curious. Are all boards that have equal white and black tiles counts solvable?

[dmurray]

Yes, by induction. Suppose the total number of squares is 2n. Remove any domino that doesn't disconnect the board, and reduce to the problem for 2n - 2.

Alternative proof: no, by example. Consider the shape made up by the X's in

  XXX
  .X.
  .X.
  XXX

(All maths proofs should be presented alongside a proof of the opposite result, to allow the reader to judge which one is likely to have slipped something past you. My favourite presentation following this rule is Mental Poker [0] by the authors of RSA.)

[0] https://people.csail.mit.edu/rivest/pubs/SRA81.pdf

[Karliss]

Slightly more interesting example which isn't disconnected is a H letter consisting of 8 blocks. 3 blocks high and 2 in the horizontal section.

[mcphage]

Nope—consider a black and white square, disconnected. But if they don’t have equal black and white counts, then they’re definitely not solvable.