Science Focus (Issue 23)

The Number Sequence That Here’s a list of numbers: 1, 11, 21, 1211, 111221. Read it to yourself. Now try and guess the next number. Here’s the next number: 312211. And the next one: 13112221. Can you guess the number after that? Th i s puz z l e was g i ven to the ver y famous mathematician John Conway by one of his students. He couldn’t guess it [1]. But the answer is really quite simple: The first number is 1. When you read it to yourself, that’s “one 1”, or 11. Read 11 to yourself: “two 1’s”, or 21. Read 21 to yourself: “one 2, one 1”, or 1211. Read 1211 to yourself: “one 1, one 2, two 1’s”, or 111221. Because it’s generated by reading aloud, Conway called this an audioactive sequence [2, 3], which is also known as a look-and-say sequence. Thi s puz z le apparent l y s tar ted at the 1977 International Mathematical Olympiad [4]. When Conway heard it from a Cambridge math student who had a friend attending the competition, and after he failed to solve it, he decided to make the problem even harder. Why? This is quite standard for mathematicians – when you make a problem harder and more general, it helps you think about how to solve all possible versions of the problem. The obvious way to make the problem harder is to start with any number you like. But one of the first things Conway noticed when starting to work on this problem was that only the digits 1, 2, and 3 “occur naturally” [1]. If you want other digits you’ll need to include them in the first number. So the “interesting” digits are 1 to 3 and we should focus on them if we want to find out more about the problem. More subtly, if you extend your example out far enough you may find something about your problem worth investigating. The next number after 13112221 is 11132 | 13211, then 311312 | 11131221, then 1321131112 | 3113112211. Look at the bars we added that split each number into two parts. If we take just the first part, 11132, and treat it as a single number, we get 311312, then 1321131112 … which are the first parts of the next few numbers. The same thing is true for the second part, 13211. From this point onward the two parts in fact never interact with each other again [1], so Conway called this a “split” and the two parts its “descendants”. He then started looking for numbers that can’t be split in this way. Although there are infinitely many of these numbers, there are exactly 92 of them which must ultimately all appear as the descendants of every possible sequence, except 22, which repeats itself [3]. Since Conway obviously missed studying chemistry in school, he called them “atoms” or “elements”. More complex numbers like 1113213211 that can be split in this way are called “compounds”. So the process of splitting compounds into elements is called “audioactive decay” [5]. Surprise! When do these splits happen? For the string 11132 | 13211, you can see that the first part 11132 ends with 2, so every step from then on ends with “some number of 2’s” and keep ending with 2 no matter what the second part does. On the other side, 13211 begins with a 1 and will continue to begin with either 1 or 3, but not 2, so it will never mess with the first part [1]. Each audioactive element is assigned to one of the first 92 elements of the periodic table, as shown in Figure 1, from hydrogen to uranium. 11132 is hafnium, and 13211 is tin. The names are assigned to resemble the real physical process of radioactive decay into lighter Element Length String The full table can be found on: 92 Uranium 1 3 91 Protactinium 2 13 90 Thorium 4 1113 . . . 1 Hydrogen 2 22 Figure 1 Lengths and strings of some Conway's elements [5]. By Peace Foo 胡適之