No longer updated!

Monday, July 28, 2008

Chess-board Chunks

Below you see a paper with a chess-board print on it. We want to cut the chess-board paper into pieces (over the lines!) such that each piece has twice as much squares of one color than of the other color (i.e. twice as much black squares as white squares or twice as much white squares as black squares).

The Question: Is this possible? Give a proof!


Answer: No, it is not possible to cut the chess-board paper into pieces such that each piece has twice as much squares of one color than of the other color.

If it would be possible, then every piece would have a number of squares divisible by 3 (because if a piece has n squares of one color and 2×n squares of the other color, it has 3×n squares in total). The total number of squares of all pieces would then also be divisible by 3. This is, however, impossible since the total number of squares on the chess-board is 64, which is not divisible by 3.

No comments: