# Conway's lesser-known results

John Horton Conway is known for many achievements: Life, the three sporadic groups in the "Conway constellation," surreal numbers, his "Look-and-Say" sequence analysis, the Conway-Schneeberger $$15$$-theorem, the Free-Will theorem—the list goes on and on.

But he was so prolific that I bet he established many less-celebrated results not so widely known. Here is one: a surprising closed billiard-ball trajectory in a regular tetrahedron:

Image from Izidor Hafner.

Q. What are other of Conway's lesser-known results?

Edit: Professor Conway passed away April 11, 2020 from complications of covid-19:

https://www.princeton.edu/news/2020/04/14/mathematician-john-horton-conway-magical-genius-known-inventing-game-life-dies-age

• The latest issue of the Monthly has a paper by Conway, Paterson and Moscow. This lovely paper was written more than 40 years back, and published in a difficult-to-find festschrift for Lenstra. On this sad day, this article brings out yet again the sheer joy of Conway. Apr 12 '20 at 2:31
• Apr 12 '20 at 2:39
• @ToddTrimble News is circulating that he passed away because of coronavirus. Apr 12 '20 at 2:39
• Coxeter and Conway constructed the beautiful "frieze patterns" in the 70s from polygonal triangulations, seemingly out of nowhere; their work was only "properly" understood many years later in the context of cluster algebras. (Probably this counts as a "well-known" result, though.) Apr 12 '20 at 3:20
• @MarkSapir You know, some and perhaps many of his results may be buried in newsgroup posts, etc., or made known to others by means other than papers. Apr 12 '20 at 4:10

The Conway immobilizer problem

The set-up:

• There are three positions (left, middle, right) on the table and there are three cards labelled $$A,B,C$$. Each card is face-up and occupies one of the three positions. A position might contain zero, one, two. or all three cards.
• Only the top card in each position is visible. If exactly two cards are visible, it is not known which of them does conceals the hidden third card.

The game:

• The goal is to have all three cards in the middle position with $$A$$ above $$B$$ above $$C$$. (As soon as the goal is reached, a bell rings.)
• A move consists in moving one card at a time from the top of one position to the top of another position.
• The crux is that the player has no memory of what he has done in the past. The player must decide his moves based only on what is currently visible.

This puzzle has been popularized by Peter Winkler, and it has been discussed in one of Winkler's books. A full solution to the puzzle can be found in:

John Horton Conway, Ben Heuer:
All solutions to the immobilizer problem.
Mathematical Intelligencer 36 (2014), no. 4, 78-86.

I believe that the theorem to which he referred to as the murder weapon has not been mentioned so far.

The murder weapon is the main theorem in a paper he coauthored with H. S. M. Coxeter and G. C. Shephard in the 1970's and whose title is "The centre of a finitely generated subgroup" (Commemoration volumes for Prof. Dr. Akitsugu Kawaguchi's seventieth birthday, Vol. II. Tensor (N.S.) 25 (1972), 405-418; erratum, ibid. (N.S.) 26 (1972), 477.).

I first read about this peculiar contribution of his in the interview of I. Hargittai with him that appeared in the March 2001 issue of The Mathematical Intelligencer (pp. 7-14). This is what Conway told Hargittai about the theorem under discussion:

Coxeter came to Cambridge and he gave a lecture, then he had this problem for which he gave proofs for selected examples, and he asked for a unified proof. I left the lecture room thinking. As I was walking through Cambridge, suddenly the idea hit me, but it hit me while I was in the middle of the road. When the idea hit me, I stopped and a large truck ran into me and bruised me considerably, and the man considerably swore at me. So I pretended that Coxeter had calculated the difficulty of this problem so precisely that he knew I would get the solution just in the middle of the road. In fact I limped back afer the accident to the meeting. Coxeter was still there, and I said, "You nearly killed me." Then I told him the solution. It eventually became a joint paper. Ever since, I've called that theorem "the murder weapon" One consequence of it is that in a group if $$a^{2} = b^{3} = c^{5} = (abc)^{-1}$$, then $$c^{610}=1$$.

You can take a look at this previous discussion in MO if you feel like hearing of some additional exploits of his...

• So did Randall Munroe get this joke from Conway? Jun 15 '20 at 11:58
• I don't know the answer to that question... Anyway, thanks for sharing that link to xkcd with me. Jun 15 '20 at 21:10

In addition to his well-known Doomsday Algorithm for calculating, he had thoughts on various other calendrical systems and dates, some of which are described here. In particular, he once told me (in person) about an algorithm that allowed him to convert between Hebrew and Gregorian dates in his head (which took him about 1.5 times as long as his Doomsday Algorithm).

I don't remember all the details, and I would love it if someone can please fill them in. I remember that there were three constants called "he", "she", and "it" that you have to memorize to do the computation.

Some brief mentions to this are found here and here, as well as p.388 of his biography.

It has been some time since I read about it and I find it difficult to find a reference for it but I recall Conway asked, if we call a triangular array of hexagons with $$n$$ hexagons along its sides $$T_n$$, which $$T_n$$ can be tiled by copies of $$T_2$$, which he named "tribones"?

The answer was some condition on $$n$$ modulo 12, proved via "Tiling Groups", but he remarkably showed that this result couldn't obtained through any colouring argument!

EDIT: It appears to be mentioned in this overview of tiling results and they provide a helpful illustration:

The condition is $$T_n$$ can be tiled iff $$n=0,2,9,11$$ modulo 12.

In "THE SEQUENCE SPACES $$l(p_\nu)$$ AND $$m(p_\nu)$$" by S. Simons there is the following theorem about sequence spaces of $$l_p$$-type with varying exponents which is attributed (without a more precise reference) to H.T. Croft and Conway:

For a sequence $$p_\nu$$ of positive numbers, $$l(p_\nu)$$ denotes the space of sequences $$(a_n)$$ such that $$\sum_\nu |a_\nu|^{p_\nu}$$ is finite and $$l_1$$ is the usual space of absolutely summable sequences.

Theorem: We suppose that $$0 < p_\nu \leq 1$$ for all $$\nu$$, and write $$\pi_\nu$$ for the conjugate index of $$p_\nu$$, i.e. $$(l/p_\nu) + > (1/\pi_\nu) = 1$$, giving $$\pi_\nu$$ the value $$-\infty$$ when $$p_\nu = > 1$$. Then the following are equivalent:

1. $$l(p_\nu) = l_1$$
2. $$\sum_\nu N^{\pi_\nu} < \infty$$ for some integer $$N > 1$$.

Conway polynomials are irreducible polynomials that provide a basis for finite fields of order $$p^n$$. While there is a unique finite field of order $$p^n$$, there are many ways of representing its elements as polynomials, all resulting in the same field (up to isomorphism). Conway polynomials provide a standardize choice of basis.

They are nice because they satisfy a certain compatibility condition with respect to subfields (i.e., fields of order $$p^m$$ with $$m$$ dividing $$n$$). Formally, the Conway polynomial for $$\mathbb{F}_{p^n}$$ is defined as the lexicographically minimal monic primitive polynomial of degree $$n$$ over $$\mathbb{F}_p$$ that is compatible with the Conway polynomials of all its subfields (see Wikipedia for more details). Of course, imposing the lexicographical ordering is a convention and is necessary to make them unique.

Conway polynomials are very useful when performing computations using computer algebra systems. They also provide portability among different systems. Computing them in general is hard, however for many small cases they have been tabulated (e.g., see the extensive tables computed by Frank Lübeck). These tables are available, for example, in Sage.