Math Party Sharing

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For the end of term math party, I shared one of my favourite problems: the mutilated chess board problem. It goes like this ...

Suppose that one domino covers exactly two adjacent squares on a chess board. If the 1x1 squares on one pair of opposite corners are removed from a standard 8x8 chess board, can the remaining squares be covered by 31 of these dominoes such that there are no gaps or overlaps (ie. tesselate the board)?

Solution:

The solution is remarkably simple, and I think that is what gives the problem its beauty.

Notice that any two adjacent squares of opposite colour. Therefore, a domino that covers exactly two adjacent squares must cover one white square and one black square.

When opposite corners are removed from the board, the removed squares are both of the same colour (either both black or both white). Thus, of the remaining 62 squares on the board, there will be 30 of one colour and 32 of the other colour. Since there are an unequal amount of squares of each colour, the dominoes will not be able to tesselate the mutilated chess board. 


Comments

  1. Susan Gerofsky31 December 2020 at 17:02

    One of the great chessboard puzzles! Thanks for sharing this, Adam.

    ReplyDelete

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