The Mutilated Chessboard

A Mutilated Chessboard

Consider a chess board, and a set of dominoes, each of which can cover exactly two squares. It's easy enough to cover the chess board completely and exactly with the dominoes, and it will (rather obviously) require exactly 32 dominoes to do so.

If you think you know everything about this already, you might want to head over to The Mutilated Chessboard Revisited ...

If we mutilate the chessboard by removing one corner square (or any single square, for that matter) then it's clearly impossible to cover it exactly, because now it would now take 31.5 dominoes.

Something not often mentioned is that cutting off two adjacent corners leaves the remaining squares coverable, and it takes 31 dominoes. It's not hard - pretty much the first attempt will succeed.

So here's the classic problem:

One of the key characteristics of mathematicians and puzzlers is that they don't simply give up, they try to prove that it's impossible.

So you try it, and the first attempt fails. Then the second, and the third, and the fourth attempts fail, and after a while you start to wonder if it's possible at all.

So that's the question - when two opposite corners have been removed from the chessboard, is it possible to cover it completely and exactly with dominoes?

Or can you prove that it's impossible?

If you've seen this, or if you solve it, or if you simply want to know more, you might like to visit this page:

• The Mutilated Chessboard Revisited

Reference:

• Martin Gardner (1994),
  • My Best Mathematical and Logic Puzzles,
    • Dover, ISBN 0-486-28152-3

Quotation from Tim Berners-Lee