“Everything in society today that can be solved by straightforward analysis gets solved immediately,” says Leonard Mlodinow, theoretical physicist and author of Elastic. Unfortunately, many problems aren’t that simple. To work one’s way through them, he says, requires a different way of thinking. It’s a creative approach familiar to mathematicians and physicists, and one that involves finding fresh ways to look at difficult problems. Mlodinow shows how this can work using the “mutilated checkerboard problem” in his Big Think+ video, “Make Progress with Elastic Thinking.”
The Mutilated Checkerboard Problem
You have an eight-by-eight checkerboard with 64 black and red squares. You also have dominoes, each of which can cover two squares horizontally or vertically. It takes 32 dominoes to cover all 64 squares.
Now, you remove the two black squares at opposite corners of the checkerboard. (This also works with the two opposite-corner red squares, but let’s use black here.) This leaves you with a “mutilated” checkerboard.
Here’s the problem/puzzle: Can you now cover the remaining 62 squares with 31 dominoes?
The straightforward answer
One way to figure this out is to try out different domino arrangements to see if it can be done. “So you start by putting down dominoes, and you get to the point where either you cover it and you say, ‘I’m done,’ or you say, ‘Whoops, it doesn’t work, I haven’t covered it. I’ll start another method and try to cover it.’” However, in that second case, when would you feel confident enough that you’d tried every possible permutation? Unless you got really lucky and hit the right layout quickly — if there is a right layout — this approach is likely to be time consuming at best.
The elastic approach
Mlodinow proposes attempting to identify the laws governing the tidy placement of dominoes on our original 64-square, “unmutilated” checkerboard. Such elastic thinking may be able to solve our 62-square problem more quickly, and more definitively.
The first, most obvious, law is that each domino covers two squares. From this, we understand that we can only cleanly cover all squares when there’s an even number of them. An odd number will leave us with a domino hanging off the edge in midair.
We’ve removed the two opposite-corner black squares, so we have 62 squares left, an even number. Are we good to go?
Nope. To fully understand the puzzle, Mlodinow says, we need to go back to our 64-square checkerboard and see if there are any other laws to be satisfied. There is one, and it so happens that it solves our problem: Each domino, whether arranged horizontally or vertically, covers one black and one red square. By removing the two corner squares, we’ve left ourselves with an uneven number of red and black squares, 32 red squares and just 30 black. This means that 31 dominoes will not cover our 62 remaining squares.
Elastic thinking for the win.