By Chris Robinson, The Blake School, @Isomorphic2CRob
Let’s begin with our assumptions:
- A square board with even side length of two squares or greater.
- Chessboard has the typical alternating color pattern.
If we remove a square of each color, there is only one rectangle, which contains those two squares as its corners.
The key concept is that this red rectangle is always even by odd or odd by even.
Imagine for a moment that the rectangle was of even width, then the top left and top right-most squares would be different colors, so the height would have to be 1, or 3, or 2n+1 to maintain opposite colored corners.
Imagine for a moment that the rectangle was of odd width, then the top corners would be the same color, so the height of the rectangle would need to be 2, 4, or 2n to ensure a different color in opposite corners.
So what’s so important about this rectangle?
We shall approach the tiling of the exterior and interior of the rectangle separately.
Once we remove the red rectangle we can divide the frame into up to 4 rectangles. Without loss of generality we shall assume that the odd-length of the red rectangle is the height. We take full vertical strips of the board on either side of the inner rectangle; these vertical strips have even height. They also leave even-width rectangles above and below the red rectangle as shown.
The fact that each of the blue border rectangles has at least one even side allows each rectangle to be easily tiled in an array.
Thus the exterior can always be tiled.
Keeping the orientation the same, we can cut vertical strips of even length and width 1 above or below the two removed squares. This leaves a rectangle with an even width as we removed one square from each side of the original square. This remaining rectangle can be tiled with an array again.
Thus the interior can always be tiled.
So with both the interior and the exterior of our red rectangle tiled, we have shown it to always be possible to tile the chessboard when one tile of each color is removed.
A sample application of this technique is shown below.