This content is locked. Please login or become a member.
Make Progress with Elastic Thinking: Solve Difficult Problems by Looking at Them Differently, with Leonard Mlodinow, Theoretical Physicist and Author, Elastic
One group of people for whom elastic thinking is a way of life is mathematicians, and also theoretical physicists. The job of a mathematician or a theoretical physicist is to find new approaches and new ideas to problems that have not been solved before. And everything in society today that can be solved by straightforward analysis gets solved immediately. Kind of like the stock market where you need inside information, you need inside thinking to make progress in math and physics today. What’s interesting is if you watch the way that mathematicians and physicists are trained to think, it’s to look at things from a different way. It’s to find new ideas and new approaches, and that’s something that’s very useful in everyone’s life. You might be in a life that you may be more or less satisfied with, but you might have a whole other potential that’s hidden from you because you’re not looking at life in a different way. You get used to looking at things in the same way.
As an illustration of how that works, there’s an interesting problem called the Mutilated Checkerboard Problem. If you have a checkerboard, it’s 8 x 8 – 8 squares by 8 squares. You have dominoes that cover 2 squares – 2 adjacent squares – either horizontal or vertical. You can cover the entire 64 squares of the checkerboard with 32 dominoes. The mutilated part comes in when I now remove the black corners at the opposite ends of the checkerboard, or the red corners at the other opposite ends. Now I have a mutilated checkerboard that has 62 squares, but the dominoes cover 2 squares. So the question is: can I cover those 62 squares with 31 dominoes?
Now, there’s different ways you can approach the problem, but the straightforward way is the way, let’s say, traditional programmers would approach it if they were programming a computer to find the answer, would be to just start trying things. So you try and you rule out. 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”. Well, that would be an extremely large combinatorial problem and it maybe could be done by a very fast computer today. But there’s another way that’s very simple, and almost trivial, if you look at it in the right way. And that’s really the key to a lot of elastic thinking, is that the thinking isn’t that hard once the looking at the problem is done right.
In this case, the way to look at the problem is to think not about physically how the dominoes cover the checkerboard, but what are the laws that govern dominoes being put down on the checkerboard? So one law is that a domino covers 2 squares. By that you know that if I had taken off just one square of the checkerboard, I couldn’t do it – I couldn’t cover it with the dominoes without something sticking out. There would be 63 squares and each domino covers 2 squares. There’s no way I can cover 63 squares with dominoes that cover 2 squares. It has to be an even number. So that would be one law.
But I took out 2 squares. So the question is: is there a law that tells you that you can’t do it? Or maybe that you can do it? What’s the answer? The answer is another law, and that’s the law that each domino doesn’t just cover 2 squares – it covers a black square and a red square. Once you realize that each domino covers a black square and red square, and you realize that the opposite ends of the checkerboard are the same color, you know it can’t be done because now we have a checkerboard of 62 squares of which 32 are black but only 30 are red. Since there is an unequal number of red and black squares, you can’t cover the checkerboard using dominoes that cover always one red square and one black square. Once you realize that, the problem is solved.