Consider these four sequences taken in succession:
Do you see the pattern of how the sequences are formed? Let’s add a few more:
The method used to create the next sequence in the list is incredibly simple: read off each digit in succession, counting the number of time it occurs, or its frequency. For example, “21” is read as “one 2, and one 1”, resulting in the sequence 1211. The last sequence above is 312211, so we read this as “one 3, one 1, two 2’s, and two 1’s”, to obtain the next sequence in the list:
To make sure you understand the sequence, check that the next terms are:
1113213211, 31131211131221, 13211311123113112211.
These sequences were discovered by John H. Conway, and he called them audioactive sequences since they come about by speaking them. For that reason, they’ve also been called look and say sequences. You can start with any integer and form other audioactive sequences. For simplicity, we only consider the ones formed by starting with 1.
For those of you who don’t know, Conway is one the world’s most famous mathematicians. He’s had so many remarkable discoveries, ranging from sporadic finite simple groups, the surreal numbers, to the Game of Life. I recommend Siobhan Robert’s biography about him entitled Genius at Play.
Properties of audioactivity
The following are some basic properties of audioactive sequences, and these can be verified by using induction on the length of the sequence.
- Their digits are either 1, 2 or 3.
- They always end in 1.
- They begin with 1 or 3, except for the third sequence 21.
- The sequence 22 is stable, in the sense that it never changes.
A remarkable thing about audioactive sequences is that they grow predictably in size. In particular, the number of digits in the (n+1)th term is about 1.3057 times the number of digits in the nth term. In particular, the ratio of two successive terms is a constant, which is now called λ or Conway’s constant.
Amazingly, λ is an algebraic number that is the unique real root of a polynomial of degree 71:
If you are startled by the complexity arising from such simple-seeming sequences, you are not alone!
Audioactive sequences split in the sense that the more complex ones break up into smaller, atomic ones. More precisely, each audioactive sequence is the concatenation of atomic audioactive ones. There are exactly 92 of these atomic sequences, and Conway named them after the 92 elements in the periodic table from hydrogen 22 through to uranium 3. The longest atomic sequence is rhenium: 111312211312113221133211322112211213322113.
The atomic audioactive sequences evolve as the sequence progresses. See the figure below capturing this evolution.
Conway recounts how he discovered audioactive sequences in the following video.
In the video, Conway aptly describes audioactive sequences as “…the stupidest problem you can conceivably imagine leading to the most complicated answer you can conceivably imagine.” That description also applies to many of the unanswered questions in mathematics.