This is one of the earliest puzzles published
in Martin Gardner's
Scientific American feature, "Mathematical Games."
|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:
Martin Gardner's original
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.
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
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