Squaring Off

Do you like to tackle problems head-on? Are you game for challenge to prove your bravado?

Here is an interesting logic puzzle. As you know, a chessboard has 8 rows and 8 columns of squares alternating between white and black. Cut off the two white corner squares of a chessboard. What we get is shown in the picture below. Originally it had 64 squares, but now there are 62 squares. Suppose you had a set of 31 rectangular dominoes, each of which is large enough to cover two squares. Can you think of a way to cover all the 62 squares with the 31 dominoes?
chessboard xx.png
If you do not have a real chessboard and dominoes, you may want to simulate doing this by using a pencil to try to draw on the diagram. After trying a few times, you might get the sense that this cannot be done. But ... how can you be so sure? Maybe you haven’t tried hard enough? Can you actually prove that it cannot be done? There are so many gazillions of combinations of ways to place the dominoes, and you need to show that every one of them does not work.

How do you do that? Are you going to outsource the problem to some people in Timbuktu to help you check all the possibilities? Are you sure you can cough up the dough even with their low wages? Or maybe you write a computer program to check every one of those possibilities?

Sometimes while squaring off with a bull, if you take the bull by its horns, you may get ... er ... liquidated. Squared off. Pun intended. OK, the direct way of tackling the problem seems futile. Even lions prefer to attack bulls from behind, not from the front. How about an indirect way? Think of the opposite. Suppose it could be done. Now what?

Notice that every domino always covers one white square and one black square. It will not cover two white squares or two black squares as it does not go diagonally. So all the dominoes would cover a total of 31 white squares and 31 black squares. But the “squared-off” chessboard has 30 white squares and 32 black squares. This does not match up (even though there is a total of 62 squares for both the dominoes and the “squared-off” chessboard). So we have a contradiction. That means it is impossible to cover the “squared-off” chessboard with the 31 dominoes. Solved!

This way of arguing is called “proof by contradiction”. If you are one of those people who like to impress with your knowledge of Latin, it is called “Reductio ad absurdum”. You reduce an assumption until you get something absurd. In other words, you assume the opposite of what you want to prove, and you find fault with it.

This is in stark contrast with the way the media, advertisers or politicians try to convince people. They just assume that they are right, and they just repeat and repeat and repeat their proposition until it infects your brain. This is not using logic, it is propaganda. This is not education, it is indoctrination.