Question

Show that by removing two white squares and two black squares from 8 X 8 checkerboard (colored as in the text) you can make it impossible to tile the remaining squares using dominoes.

Short answer

It is not possible to use dominoes to tile a 8 X 8 checkerboard when two black squares and two white squares are removed.

Step-by-step solution

To prove

When two black squares and two white squares are removed from a 8 X 8 checkerboard, it is impossible to tile the remaining squares using dominoes.

Proof

Rotate the 8 X 8 checkerboard such that a black square is in the upper left corner.

Then, remove the two squares adjacent to the upper left corner and remove the two black squares adjacent to the upper right corner.

Now, it is impossible to tile the 8 X 9 checkerboard because we cannot tile the

Upper right corner nor the upper left corner.

Thus, it is not possible to use dominoes to tile a 8 X 8 checkerboard when two black squares and two white squares are removed.