The Number Sequence That Needs to Be Said
By Peace Foo 胡適之
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?
This puzzle was given to the very 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.
This puzzle apparently started 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.
| Element | Length | String |
| 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]. (The full table can be found here.)
The names are assigned to resemble the real physical process of radioactive decay into lighter elements. For example, uranium (3) decays into protactinium (13), then into thorium (1113), and so on. One might expect a lighter element to have a longer string under audioactive decay. However, sometimes an element also decays into a combination of “shorter” elements, which is why some elements are shorter than their predecessors [3] – for instance, promethium (132) follows samarium (311332). The next number after 311332 should be 13212312, but it can split into three lighter elements, promethium (132), calcium (12) and zinc (312). In addition, the original starting point of 1, 11, 21, 1211 until 13112221 are referred to as “primordial elements” because they can’t be split either but are not involved in every possible sequence [5]. Conway focused on these 92 numbers, again, because they are “sufficiently general” mathematically; this means they can tell us something interesting about the sequence no matter which starting point we have.
It should be possible for you to guess that since all elements occur in a decay process, any possible decay process will eventually result in only these 92 elements. But as all mathematicians know, a guess is not enough. Conway proved this result over about a month of work with assistance from a fellow mathematician and called it the “Cosmological Theorem” [3]. Soon later another simpler proof was announced, but unfortunately both proofs were not published. Eventually the result was proven again by several others [6].
A consequence of this theorem is that the number of digits of successive numbers increases at a constant rate for all sequences [3]. In our original sequence starting from 1, these lengths are 1, 2, 2, 4, 6, 6, 8, 10, 14, 20, … which you can calculate for yourself gradually approximates toward a ratio of 1.303577… times the number of digits in the previous term [1]. Based on the proof of the Cosmological Theorem, some application of linear algebra can tell you that this ratio is a solution to some equation of degree at most 92 (i.e. the highest power of the unknown is x92) [3]. Conway and his colleagues then deduced that 1.303577… is actually the largest real root of a specific 71-degree polynomial so unnecessarily complex that we’re afraid to print it here [2].
Why does the answer to such a simple sequence involve monstrous decay chains, matrices and 71-degree equations? We don’t know, but it shows how even the simplest questions can produce genuinely engaging mathematics if you know how to look for it. Mathematics is a sport; mathematicians love to challenge themselves. Challenge yourself enough, ask your questions in the right way, and like Aladdin’s cave the door to a whole world of interesting new insights will open.
References:
[1] Numberphile. (2014, August 8). Look-and-Say Numbers (feat John Conway) – Numberphile [Video file]. Retrieved from https://www.youtube.com/watch?v=ea7lJkEhytA
[2] Bonato, A. (2018, May 2). Audioactive Sequences. Retrieved from https://anthonybonato.com/2018/05/02/audioactive-sequences/
[3] Conway, J. H. (1987). The Weird and Wonderful Chemistry of Audioactive Decay. In T. M. Cover, & B. Gopinath (Eds.), Open Problems in Communication and Computation (pp. 173–188). New York, NY: Springer. doi:10.1007/978-1-4612-4808-8_53
[4] Mowbray, M., Pennington, R., & Welbourne, E. (1986). Prelude. Eureka, (46), 4. Retrieved from https://www.archim.org.uk/eureka/archive/Eureka-46.pdf
[5] Hilgemeier, M. (1993). Audioactive Decay. Retrieved from http://www.se16.info/mhi/Part1.htm
[6] Ekhad, S. B., & Zeilberger, D. (1997). Proof of Conway’s Lost Cosmological Theorem. Electronic Research Announcements of the American Mathematical Society, 3, 78–82.