Prove with a contradiction proof that an 8x8 checkerboard wi

Prove with a contradiction proof that an 8x8 checkerboard with the upper right corner removed cannot be covered by 1x3 tiles. (Hint: consider three colors). Is a covering possible if the upper left corner is removed instead ?

Solution

Solution:

Cotradiction Proof:

Color the 8x8 checkerboard, starting with the top left square with the colors blue,yellow,red repeating this sequence (from top to bottom and left to right) until all squares are covered.

Now remove the top right square (yellow) from the checkerboard.

Assume by contradiction that it is possible to cover this board with 1x3 tiles.

In any covering ,any tile covers exactly one blue, one yellow and one red square, so there must be an equal number of squares colored blue,yellow and red;

but there are 22 squares colored blue, 20 colored yellow and 21 colored red, which is contradiction.

If the upper left corner is removed the board still cannot be covered by 1x3 tiles: by contradiction suppose it is possible to cover such a board; by rotating 90 degrees clockwise we can produce a covering of an 8x8 board with the upper right square removed, and this is impossible.