Professor Stewart's Hoard of Mathematical Treasures (6 page)

You’ve probably heard of magic squares - grids of numbers that add up to the same total when read horizontally, vertically or diagonally. Magic hexagons are similar, but now the grid is a honeycomb, and the three natural directions to read the numbers are at 120° to each other. In Cabinet (page 270) I told you that there are only two possible magic hexagons, ignoring symmetrically related ones: a silly one of size 1 and a sensible one of size 3.

The only possible normal magic hexagons, of size 1 and 3, and an abnormal hexagon of size 7.

That’s true for ‘normal’ magic hexagons, where the numbers are consecutive integers starting 1, 2, 3, . . . . But it turns out that there are more possibilities if you allow ‘abnormal’ ones, where the numbers remain consecutive but start further along, say 3, 4, 5, . . . . The largest known abnormal magic hexagon was found by Zahray Arsen in 2006. It has size 7, the numbers run from 2 to 128, and the magic constant - the sum of the numbers in any row or slanting line - is 635. Arsen has also discovered abnormal magic hexagons of size 4 and 5. See
en.wikipedia.org/wiki/Magic_hexagon

Simple questions need not be easy to answer. Here’s a famous example. You can explore it with pencil and paper, or a calculator, but what it does in general baffles even the world’s greatest mathematicians. They think they know the answer, but no one can prove it. It goes like this.

Think of a number. Now apply the following rules over and over again:

What happens?

I thought of 11. This is odd, so the next number is 3×11 + 1 = 34. That’s even, so I divide by 2 to get 17. This is odd, and leads to 52. After that the numbers go 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. From there, we get 4, 2, 1, 4, 2, 1 indefinitely. So usually we add a third rule:

In 1937, Lothar Collatz asked whether this procedure always reaches 1, no matter what number you start with. More than seventy years later, we still don’t know the answer. There are several other names for this problem: the Syracuse problem, the 3
n
+ 1 problem, the Ulam problem. It is often posed as a conjecture which states that the answer is yes, and that’s what most mathematicians expect.

The fates of the numbers 1-20, and anything else they lead to.

One thing that makes the Collatz-Syracuse-Ulam problem or conjecture hard is that the numbers don’t always get smaller as you proceed. The chain starting with 15 gets up to 160 before eventually subsiding. Little old 27 positively explodes:

It takes 111 steps to get to 1. But it does get there eventually.

This kind of thing makes you wonder whether there might be some particular number for which the process is even more explosive, and heads off to infinity. The numbers will go up and down a lot, of course. Any odd number leads to an increase, but the number can’t increase twice in succession: when n is odd, 3n + 1 is even, so the next step after that is to divide by 2. But the result at that stage is still bigger than n; in fact, it’s
(3
n
+ 1). However, if this is also even, we get to something smaller than n, namely
(3
n
+ 1). So what happens is quite delicate.

If no number explodes to infinity, the other possibility is that there might be some other cycle which some numbers hit instead of 4 → 2 → 1. It has been proved that any such cycle must contain at least 35,400 terms.

Up to 100 million, the number that takes longest to reach 1 is 63,728,127, which requires 949 steps.

Computer calculations show that every starting number up to at least 19×2
⁵⁸
≈ 5.48×10
¹⁸
eventually hits 1. This is impressively large, and a lot of theoretical input has to go into the computation - you don’t just check the numbers one by one. But the example of Skewes’ number (see page 46) shows that 10
¹⁸
isn’t really very big, as such things go, so the computer evidence
isn’t as convincing as it might seem. Everything we know about the question conspires to indicate that if there is an exceptional number that doesn’t hit 1, it will be absolutely gigantic.

Probability calculations suggest that the probability that some number heads off to infinity is zero. However, such calculations are not rigorous, because the numbers arising are not truly random. Exceptions might still occur anyway, and even if the argument were rigorous it would not rule out running into a different cycle.

If the process is extended so that it can start with zero or negative integers, four other cycles appear. They all involve numbers bigger than -20, so you might like to search for them (see the answer on page 279). The conjecture now becomes: these five cycles are all that can happen.

There are also connections with chaotic dynamics and fractal geometry, which lead to some beautiful ideas and pictures, but don’t solve the problem either. There’s a lot of information about this problem on the internet, for example:

Rattler’s Jewellers had promised Mrs Jones that they would fit her nine pieces of gold chain together to make a necklace, an endless loop of chain. It would cost them £1 to cut each link, and £2 to rejoin it - a total of £3 per link. If they cut one link at the end of each separate piece, linking the pieces one at a time, the total cost would be £27. However, they had promised to do this for less than the cost of a new chain, which was £26. Help Rattler’s avoid losing money - and, more importantly, make the cost to Mrs Jones as small as possible - by finding a better way to fit the pieces of chain together.

Nine lengths of chain.

Answer on page 279

Our first cat, who rejoiced in the name Seamus Android, was possibly one of the few cats on earth that did not always land on its feet. Seamus didn’t have a clue. He would come down the stairs one step at a time, head first. At one point, Avril tried to train him to land on his feet by holding him upside down over a thick cushion and letting go. He liked the game but made no effort to turn in mid-air.

Oops . . . What do I do now?

There is a mathematical issue here. Associated with any moving body is a quantity called angular momentum, which, roughly speaking, is the mass multiplied by rate of spin about a suitable axis. Newton’s laws of motion imply that the angular momentum of any moving body is conserved, that is, does not change. So how can a falling cat turn over without touching anything?

 

Answer on page 279

A cat is not the only proverbial falling object. Toast is another. It always lands buttered-side down. If not, you must have buttered the wrong side.