Cover it with dominos
mathematical insight is powerful
...and is fun!

This is one of the earliest puzzles published in Martin Gardner's
Scientific American feature, "Mathematical Games."
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 Start with a checkerboard and 32 dominoes.  Each domino is of such size that exactly covers two adjacent squares on the board.  The 32 dominoes therefore can cover all 64 of the chess board squares.  But now suppose we cut off two squares at diagonally opposite corners of the board and discard one of the dominoes.  Is it possible to place the 31 dominoes on the board so that all the remaining 62 squares are covered?  If not prove it impossible. There were two different versions of the checkerboard this year:
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 Subtle Puzzle Martin Gardner's original Simple Puzzle Our modified version
 Puzzle solvers quickly discovered that with the modified version it's very easy to place the dominos and with the original version it's very difficult--impossible, in fact. This puzzle illustrates how mathematical insight can give us surprising and powerful knowledge.  Let's modify the original puzzle in a different way.  This time start not with a checkerboard but with the same pattern without the black and white distinctions: The 64 squares can be covered with 32 dominos, but if we remove two of the squares we might or might not be able to cover the remaining 62 squares with 31 dominos.  There are 64 X 63 / 2 = 2016 different ways we can remove two squares.  With half of those ways we can cover them with the 31 dominos, and with the other half we cannot. Coloring the squares like a checkerboard doesn't make any difference to how the dominos can or cannot cover squares, but it does help us see why we might not be able to cover them, and it let's us easily identify which half of the 2016 is which.  The mathematical insight is simply that a domino covers one white square and one black square on the checkerboard, and if we remove two blacks or two whites we cannot put the 31 dominos down to cover the remaining 62 squares. Discover this fact in a most unusual way? That's the kind of simple and "elegant" insight that makes mathematicians always want more and more mathematics.  It's the real mathematics that Keith Devlin describes in his 2002 book, The Math Gene. Everyone can do math, and everyone can enjoy math, when they discover what it really is. Just do it!

Previous year's looks at elegant puzzles

Scope -- Relevance -- Math -- Logic -- Tensor