Wednesday, June 3, 2015

A Checkerboard Problem


Max Black proposed this problem in 1945.  He was a British philosopher and mathematician.

Suppose you had a chess or checker board (8 x 8 = 64 squares), but you cut off a corner square from two opposite corners.  You would get a board that looked like the picture above, with just 62 squares.
Now suppose that you had 31 dominoes.  Each domino was just the right size to cover two squares on the board.  31 dominoes, each covering 2 squares would cover a total of 62 squares (31 x 2 = 62).
Can the 31 dominoes be arranged on the board to cover all 62 squares?
When I first saw this problem I felt that the answer was probably “Yes, it is possible.”  I thought I could just make two copies of the 62 square board, and cut one of them into two square “dominoes” and arrange them on the second copy of the board.
Try it.  If you try it you will probably learn what I learned.

I’ll post a follow up in a few days to see how you did.


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