Letters to a Young Mathematician by Ian Stewart

Author:Ian Stewart [Stewart, Ian]
Language: eng
Format: epub, mobi, pdf
Tags: Non-Fiction, Philosophy, Science
ISBN: 9780465008414
Publisher: Basic Books
Published: 2006-03-27T00:00:00+00:00

13

Impossible Problems

Dear Meg,

Please don’t try to trisect the angle. I’ll send you some interesting problems to work on if you want to flex your research muscles already; just stay clear of angle trisection. Why? Because you’ll be wasting your time. Methods that go beyond the traditional unmarked ruler and compass are well known; methods that do not cannot possibly be correct. We know that because mathematics enjoys a privilege that is denied to most other walks of life. In mathematics, we can prove that something is impossible.

In most walks of life, “impossible” may mean anything from “I can’t be bothered” to “No one knows how to do it” to “Those in charge will never agree.” The science fiction writer Arthur C. Clarke famously wrote that “When an elderly and distinguished scientist declares something to be possible, he is very likely right. When he declares it to be impossible, he is almost certainly wrong.” (Clarke was writing in 1963, when most scientists, especially elderly and distinguished ones, were almost certainly “he.”) But applied to mathematicians and mathematical theorems, Clarke’s statement is plain wrong. A mathematical proof of impossibility is a virtually unbreakable guarantee.

I say “virtually” because sometimes the impossible can become possible if the question is subtly changed. Then, of course, it’s not the same question. Archimedes knew how to trisect an angle using a marked ruler and compass.

My favorite simple impossible problem is a puzzle. Though it looks frivolous at first glance, it provides a lot of insight into logical inference in mathematics, and in particular into how we know that some tasks are impossible. The puzzle is this: given a chessboard with two diagonally opposite corners missing, can you cover it with thirty-one dominoes, each just the right size to cover two adjacent squares?

It must be understood that no “cheating” is allowed. The dominoes must not overlap, or be cut up, or anything like that.

The first question to ask is reasonably straightforward and comes naturally to any mathematician: is the area an obstacle to success? The total area of the mutilated chessboard is 64 - 2 = 62 squares. The total area of the dominoes is 2 × 31 = 62 squares. So we have exactly the right number of dominoes to cover the chessboard. If we had been given only thirty, then calculating the total area would immediately have proved that the task is impossible. But we’ve been given thirty-one, so the area is not an obstacle.

Meg, I know you’ve done a lot of math, but it’s just possible that you’ve never come across this puzzle. Puzzles do not feature prominently in university textbooks. Please try it. For the moment, don’t think about it; just cut out some cardboard dominoes and try to fit them together.

Done that? Did you get anywhere?

No. You tried and tried but nothing worked. You can see why if you count the black and white squares.

Every domino, no matter where it is placed, covers one black and one white square of the chessboard. So any arrangement of nonoverlapping dominoes must cover the same number of black squares as white.

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