The Sequence That Reads Itself Aloud

Write down a row of digits. Now describe it out loud, going left to right, by saying how many of each digit you see in a row: not what the number is, but what it looks like. Write down what you said, using digits, and then describe that.

Start with 1. That is one 1, so write 11. That is two 1s, so write 21. That is one 2 and one 1, so write 1211. That is one 1, one 2 and two 1s, so write 111221. That is three 1s, two 2s and one 1, so write 312211.

The important detail is the word "in a row". In 111221 the three leading 1s are described once, as a count of three, and the single 1 at the end is described separately because two 2s stand between them. A block of the same digit with something different on each side is called a run, and the rule is: replace every run by its length followed by its digit.

Two things go wrong if you expect this to behave like arithmetic, and both are worth seeing early. Start from 22 and nothing happens at all: two 2s is written 22, which is where you began, so the sequence stands still forever. Start from 5 and every term for as long as you care to compute ends in a 5, because reading a run of 5s aloud puts a 5 straight back.

Two questions are worth separating, because the answers are of very different kinds.

Why can a digit never disappear? This one is easy and complete. If the digit $d$ appears anywhere in a term, it sits inside some run of $d$s, of some length $c$. The rule replaces that run by the two digits $c$ and $d$. So $d$ is still there in the next term, and by the same argument in the one after that, forever. That single line settles what the seed 5 is doing, and it also explains why the famous statement about this sequence has to name its starting point: "no digit above 3" cannot possibly be a fact about the rule, because the rule has no way to remove the 7 you handed it.

Why should the length grow at a steady rate? This one is genuinely hard, and the honest intuition is only half an answer. Each run in a term becomes exactly two digits in the next one, so the next length is twice the number of runs. If the runs kept a stable average length, the growth factor would be exactly 2 divided by that average, and long stretches of these terms do look statistically similar to one another: mostly runs of one and two, occasionally three, never longer from the seed 1. Growth by a roughly fixed factor is what you would expect from a rule that keeps producing the same mixture of runs.

What that argument does not do is say what the factor is, or prove there is one. The mixture of runs would have to be shown to settle, and settling is exactly the hard part. Conway's proof proceeds quite differently: past a certain point every term splits into blocks that never interact again, there are 92 of them, and the whole sequence is bookkeeping over a finite list of pieces. The growth factor falls out as the dominant eigenvalue of the matrix saying which pieces produce which. None of that happens in this browser.

The sentence worth carrying away is this one.

The rule cannot remove a digit, so what a sequence contains is decided entirely by its seed. How fast it grows is decided by nothing you can see, and is a theorem.

Three statements live on this page and they were established in three completely different ways.

• A digit that appears once appears forever. Proved here, in one line, for every seed and every step. It is the only unbounded statement on the page that this laboratory really establishes.
• From the seed 1, no digit above 3 ever appears. A theorem of Conway's, cited. The laboratory computes forty-five terms and finds the largest digit in all of them is 3. That is a fact about forty-five terms, and it would look exactly the same whether or not a 4 turned up at term 400.
• The lengths grow by a factor approaching about 1.303577. Also Conway's, also cited, and this one is worth being especially careful about, because the page prints numbers that look like measurements of it and are not.

That last point is the whole reason the growth view exists. Print the ratio of each length to the one before it and you get 1.336 at term 20, 1.3085 at term 30, and 1.3042 at term 40. Those are close to the constant, and they are visibly still moving: they wobble between roughly 1.302 and 1.306 term after term without settling anywhere. Forty terms is nowhere near enough to see a limit, and reading three matching decimal places out of a computation that supports fewer is the exact mistake this site exists to avoid. The constant is in the ledger because somebody proved it, not because this page measured it.

For a non-empty string $s$ over the digits $\{0,\dots ,9\}$, write $s=d_1^{c_1}{d}_2^{c_2}\cdots d_r^{c_r}$ for its decomposition into maximal runs, so $d_i\ne d_{i+1}$ and every $c_i\ge 1$. Define the look-and-say map

where $\overline{c}$ is the decimal representation of $c$. Write $L^k$ for $k$ applications and ${\ell }_k(s)=|L^k(s)|$ for the length of the $k$th term.

1. If the digit $d$ occurs in $s$, then $d$ occurs in $L(s)$. Proved.
2. $L(22)=22$, and $22$ is the only $s$ with $|s|\le 5$ satisfying $L(s)=s$. Proved: the first by direct evaluation, the second by checking all $111,110$ such strings.
3. For $s=1$, every term $L^k(1)$ contains no digit greater than $3$. Proved by Conway, and cited rather than reproduced. This laboratory verifies it for $k\le 44$ only.
4. ${\ell }_k(1)$ for $k=0,\dots ,39$ is $1,2,2,4,6,6,8,10,14,20,\dots ,63138$. Exhaustively checked over exactly those forty terms.
5. For every seed $s\ne 22$, ${\ell }_{k+1}(s)/{\ell }_k(s)\to \lambda$ as $k\to \infty$, where $\lambda \approx 1.303577269$ is the largest real root of a particular degree-71 polynomial. Proved by Conway (the Cosmological Theorem), and cited. No limit is computed on this page, and the ratios it prints at $k=40$ still differ from $\lambda$ in the third decimal.

Statement 5 is the one that makes the exploration. Statements 1, 2 and 4 are what this page can establish on its own, and their agreement with 3 and 5 is consistency rather than evidence.

A digit that appears once appears forever. Let $d$ occur in $s$. Then $d=d_i$ for some run index $i$, and by definition $L(s)$ contains the substring $\overline{c_i}{d}_i$, so $d$ occurs in $L(s)$. Induction on $k$ gives the statement for every later term. $■$

Two consequences follow immediately and are worth stating, because between them they account for every failure preset on this page. The set of digits occurring in $L^k(s)$ is non-decreasing in $k$; and if the seed contains a digit above 3, so does every term after it, whatever Conway's theorem says about the seed 1.

$22$ is fixed. The string $22$ is a single run: $d_1=2$, $c_1=2$. So $L(22)=\overline{2}2=22$. The sequence from this seed is constant, which is why the growth theorem in layer 4 has to exclude it. $■$

$22$ is the only fixed string of length at most 5. Each of the $10+10^2+10^3+10^4+10^5=111,110$ strings of one to five digits, leading zeroes included, was mapped once and compared with itself by exact string equality. Exactly one matched. The space is finite and every element of it was checked, so this is a proof about strings of length at most 5, and it says nothing whatever about longer ones. $■$

One structural fact narrows the search, and is worth having even though this page does not push it to a general theorem. Every run contributes exactly two digits to the reading, so a string of $r$ runs has $|L(s)|=2r$. A fixed string therefore has even length $2r$ and exactly $r$ runs, meaning its runs average length two. At length 2 that settles the matter by hand: a string of two different digits has two runs and so reads as four digits; a string $dd$ has one run and reads as $2d$, which equals $dd$ only when $d=2$.

What is not proved here. Both headline theorems. The absence of digits above 3 from the seed 1 is a statement about infinitely many terms, and forty-five of them establish nothing about the forty-sixth. The growth constant is a statement about a limit, and the ratios this page prints do not converge over the range it can compute: they are $1.3050$, $1.3032$, $1.3025$, $1.3028$ and $1.3042$ over the last five of forty terms, moving in the third decimal place in both directions. Conway's proof of both goes through the decomposition into 92 elements and the eigenvalues of the resulting matrix, and reproducing it is not something a browser can be talked into doing.

What this laboratory computes. Terms 1 to 45 from a seed of up to twenty digits, generated by exact string operations with no arithmetic in the generation at all. Forty-five is where it stops because the terms grow by about thirty per cent each time: term 45 from the seed 1 has 237,746 digits, term 60 would have tens of millions, and there is nothing to see in either of them that term 40 does not already show. The tile figure gives up well before that and prints the digits plainly instead, because a row of sixty thousand tiles is not a figure.

Ratios are not limits. The single most misleading thing this page could do is print 1.3042 next to 1.303577 and let a reader draw the obvious conclusion. Those two numbers agree to three decimal places by coincidence of scale rather than by convergence: the successive ratios are still moving in that decimal place term after term, in both directions, with no sign of settling. Raising the term count does not fix this in any way a reader can watch, since the wobble shrinks far more slowly than the terms grow.

Where the mathematics is still moving. The sequence itself is closed: Conway's Cosmological Theorem is proved, and the original proof having been lost, it has since been reproved by others. The neighbouring questions are not so settled. The same rule in other bases has its own constant, and the general behaviour across bases is less thoroughly described than the base-10 story. Nothing on this page computes or claims anything about any base other than ten.

Read this next to Ducci sequences. There the whole character of the machine is decided by the number of places in the ring, and the values barely matter. Here it is decided by which digits the seed happens to contain, since the rule can never remove one. Both are rules whose outcome is settled by a feature of the input that has nothing to do with how large it is.