Word Play

Let’s create a sequence starting with the number 1. For clarity, we will not include the grammatically correct period at the end:

1

What do you observe? I see “one one,” indicating one occurrence of the number 1. Now, let’s convert that back into numbers:

11

In our new digit string, we now see “two ones.” Ignoring plural endings, we represent this as:

21

This leads us to:

1211

Before we continue, feel free to try writing the next iteration—it's quite enjoyable.

Starting from the beginning, the sequence develops as follows:

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ...

You may notice that “3” appears to be the largest digit in the sequence. Indeed, regardless of the starting number, no digit greater than 3 can emerge from the process after the second term.

“Interesting,” you might think. Conway first encountered this sequence at a party when a student shared a few terms and challenged him to identify the next. Initially, he couldn’t, but once it was explained, it fascinated him. This deceptively simple sequence proved to be akin to the “harmless little bunny rabbit” from Monty Python and the Holy Grail. As Conway described:

> “It seemed to me the simplest problem you could imagine that results in the most complex answer conceivable.”

Conway’s Constant

To begin with, the length of the output grows exponentially according to a value known as Conway’s constant ?, where

? ? 1.3035772690342963912…

You might find it intriguing that such an apparently random output grows in a predictable manner. Moreover, Conway demonstrated that ? is an algebraic number. Specifically, he proved it is the unique positive real root of a degree-71 polynomial! Here’s what that looks like:

In Conway’s own words:

> “In my opinion, ? holds the record for the most complicated algebraic number derived from the simplest source.”

“Fascinating,” you might say. It remains unclear why this quirky sequence is related to the root of any polynomial, much less one with numerous terms.

Audioactive Decay & The Cosmological Theorem

Now things become truly peculiar. In 1986, Conway published a paper titled “The Weird and Wonderful Chemistry of Audioactive Decay” in the magazine Eureka. To grasp the source of the exotic polynomial, we must discuss the “weird” aspect.

In the article, Conway investigates the sequence's behavior and discovers that any finite string, starting from the eighth term, will divide (“decay”) into specific, independent strings. Remarkably, there are exactly 92 of these core strings, which he named after the elements in the periodic table.

The splitting is best illustrated through an example. Here’s the eighth term in the sequence:

1113213211

We can divide it as follows:

11132.13211

From here, the strings on either side of the period will develop independently.

To clarify this concept, refer to the table below. You will see that the strings on the left side of the split continue to evolve without regard to the strings on the right. When combined, the two parts form the complete string, as if it had never been split.

Recognizing this behavior, Conway sought common atoms or elements that appear in all descendant strings. Essentially, all descendants then become compounds of these fundamental units.

Thus, he formulated his Cosmological Theorem. Broadly, it asserts that for any starting number in the look-and-say sequence, the resulting string will eventually split into a compound formed from the fundamental elements in a finite amount of time.

Here are the initial elements in Conway’s “Periodic Table”:

In a straightforward case, Term 9 in the preceding table splits into:

Element 71 (Lutetium): 311312, and Element 49 (Indium): 11131221, thus yielding:

31131211131221 = Lu.In

However, the expansion of the table quickly becomes complex. For reference, here’s the complete table extracted from Conway’s paper. As he noted:

> “It’s also true (but ASTONISHINGLY hard to prove) that every string eventually decays into a compound of these elements, along with possibly a few others (namely, isotopes of Plutonium and Neptunium as defined below).”

According to him, the original proofs he and his colleagues developed were lost:

> “Proof of the Cosmological Theorem would occupy the remainder of Eureka! Richard Parker and I discovered a proof over a month of intensive work (or, rather, play!). We first devised a complicated argument that (almost) reduced the problem to tracking a few hundred cases, then handled these on dozens of sheets of paper (now lost). Mike Guy found a simpler proof using tracking and backtracking in roughly equal parts. Guy’s proof, while still lengthy (almost entirely lost), had the advantage of identifying the longest-lived of the exotic elements, namely, the isotopes of Methuselum (2233322211n; see Figure 2). Can you produce a proof in just a few pages? Please!”

Subsequent computer-based proofs were developed by Ekhad and Zeilberger in 1997, Litherland in 2003, and Watkins in 2006. These proofs were significantly longer than “a few pages,” leaving room for further exploration if you’re interested!

The Polynomial

Regarding that substantial polynomial…

If we know the lengths of all element strings, we can calculate the lengths of all descendant strings. For example, Lutetium has a length of 6 digits, and Indium has 8 digits, leading us to conclude that Lu.In is 14 digits long.

With this knowledge, we can regard each string in the look-and-say sequence as a vector v. For a given constituent element (substring) i, we denote its length by l? and its frequency of occurrence by f?. Thus, the vector component v? equals l? × f? for each constituent substring. The total length of that string in the look-and-say sequence is the sum of all v?.

Knowing how the strings evolve with respect to the lengths of their constituent descendants allows us to construct a 92x92 transition matrix T that encodes the transformation between vectors. The sum of the vector components in Tv (the transformation of v by T) provides the length of the next string in the look-and-say sequence. Similarly, the length of the following string is the sum of the vector components for T²v, and this pattern continues for all T?v.

In essence, we have an order-92 linear recurrence relation that describes the lengths of the strings in the sequence. As this relation relies solely on preceding values, it is termed a homogeneous linear recurrence. This allows us to derive a closed-form expression using the eigenvalues of T. If you possess a computer algebra system (CAS), you can compute the eigenvalues of T using this transition matrix provided by Nathaniel Johnston at Mount Allison University.

We discover that 21 of the eigenvalues are either 0 or 1 and thus contribute nothing to the system's evolution. This observation, which Conway refers to as a “21-dimensional invariant subspace,” reduces the polynomial’s degree to 71.

The 71 eigenvalues of T that are not 0 or 1 correspond precisely to the roots of the polynomial. By factoring the degree-92 characteristic polynomial of T using a CAS, we can confirm that it reduces to a degree-71 polynomial.

We are particularly interested in the maximum absolute value of the eigenvalues of T (often termed the spectral radius of a matrix). This is the value that most significantly influences the vector during iterations.

That maximum value is, indeed, the unique positive real root (illustrated above) of our degree-71 polynomial — Conway’s ?.

“From a spoken number sequence to the periodic table of elements, culminating in the roots of a massive polynomial — that’s astonishing!” you might exclaim. I would wholeheartedly agree.

Thank you for reading! If you found this piece enjoyable, feel free to click the “Applause” icon at the top as many times as you like. You can also subscribe to receive my latest content directly in your inbox.

Further Resources

• Online look-and-say sequence generator: https://catonmat.net/tools/generate-look-and-say-sequence
• John Conway’s original article in Eureka: https://www.archim.org.uk/eureka/archive/Eureka-46.pdf
• Nathaniel Johnston’s insightful explanation of Conway’s constant: https://njohnston.ca/2010/10/a-derivation-of-conways-degree-71-look-and-say-polynomial/