Friday Puzzler (cont'd) -- Chess Board 2 of 3

Re:  Chess Board 2 of 3 partial solution

Alex Y

There is no "aha!" solution to this one. In fact, it is always possible to tile the chessboard with one black and one white square missing.

The proof involves finding a strategy for tiling the board that is easily adjustable when one white and one black square are removed. There are probably lots of such strategies. I'll leave this until the end of the week and then show the strategy I have seen (not found myself) for this.

Re:  Chess Board 2 of 3 Solution

Alex Y

Here is just one of probably very many possible strategies for tiling a chessboard that can be easily modified if a white and black (or blue and yellow in this rendering) square is eliminated.

Note that the path shown below is a complete circuit, ending up wherever you start. And it doesn't matter whether you start blue-yellow or yellow-blue, or which direction.

I've picked one blue and one yellow square to eliminate, but which ones clearly doesn't matter. If you start to the right of the missing blue square, you will put down dominoes on yellow-blue pairs along this path ending with the yellow-blue that is next to the missing yellow square. Going the other direction from the missing blue square, you tile the rest of the board, ending up next to the missing yellow square.