Some years ago, at the MathsJam annual gathering, David Singmaster ambled up to one of the other delegates. “Nice T-shirt,” he said. The design showed the small robot from Star Wars, loyal companion to C-3PO and widely adored for his playful beeps and boops. But instead of his usual cylindrical body, in its place was a Rubik’s cube. Printed underneath was the name: R2-D2.
“Yes!” said the delegate, clearly delighted that someone had noticed, “But it’s even cleverer than you might think. See, the robot’s name is R2-D2, but that’s also a move in the notation we use for solving the cube!”
David chatted with them for a while, during which time the truth slowly emerged. The notation in question is called the Singmaster notation, invented by David in late 1978 during his process of devising a method of solving (or restoring) the cube. So the delight increased, to meet the man who’d brought a method of solving the cube to a world of enthusiasts.
It’s via the cube that most people will have encountered David, but if you met him in person it rapidly became clear that his interests extended far beyond that one mathematical object. I don’t remember when I first met him, but I remember spending some time with him at the Gathering 4 Gardner in 2008. He had a new (to him, and to me) toy that we call ‘the ring on a chain’, and he was both practising how to do it, and delightedly showing it to anyone in range, sometimes getting it right.
David at Gathering 4 Gardner 13 (G4G13) in 2018
This was the theme… David had a delight and a passion for sharing. A visit to his house was an adventure, and once you were there it was extraordinarily difficult to leave! “Just one more thing” was his catchphrase, as he would first show, then watch you struggle with, then struggle in turn to remember how to solve a mechanical puzzle of one sort or another. His tongue would poke out, first on one side, then the other, then back again, all the while as he talked about the history of the object at hand.
He delighted in the struggle, and each time he succeeded in solving a puzzle again his joy was evident, and you were invited to share it with him.
There was much more to him than just puzzles. As an undergraduate, he solved a prize problem in his number theory course, and another problem that eventually led to two academic papers. He taught in Beirut, then lived in Cyprus, and later while working as a photographer for an underwater archaeological expedition off the coast of Sicily, he accidentally discovered the oldest known warship wreck in existence. He then worked in Pisa for a year, before finally settling in London.
David is known for Singmaster’s conjecture. In the pattern we know as Pascal’s triangle, every number greater than one will appear somewhere, and each will appear only finitely many times, but David noticed that none of them seemed to turn up very often. He conjectured that there is some fixed finite bound on how often a number can appear, a conjecture that remains unresolved. So far, it has been found that the number 3003 appears eight times… no other number is known to turn up as often, and while there are many related questions, almost all remain open.
John Railing, John Conway, Richard Guy and David at Gathering 4 Gardner 12 (G4G12) in 2016
David’s knowledge of the history of puzzles (among many other things) was extensive, and so was his network of connections. He was friends with John Conway, Richard Guy, Martin Gardner, and many others, and he was right at home with them, as he was with anyone else who loved puzzles, games, maths, and toys. He was kind and welcoming to all who would engage.
David was also quick and sharp. On more than one occasion, at various gatherings such as MathsJam or the Gathering 4 Gardner, a puzzle would be proposed and he would give an answer within seconds. Sometimes it was because it was similar to a puzzle he’d seen before—and it felt like he had seen them all! But sometimes it was simply because he had solved it there, on the spot. This could make him challenging company, but if you were willing to go along for the ride it was immensely rewarding, for he was free with his knowledge, and his enthusiasm was undeniable and irresistibly infectious.
Just part of David’s collection
He was also an avid collector. His book collection has something in the vicinity of 10,000 books, including works on recreational maths, the history of maths, plus a range of cartoons, humour, and language. His collection of mechanical puzzles is extensive, including (of course) many, many examples of, and variants on, the Rubik’s cube. Having written the first book outlining a method of restoring the cube, and devising the notation that is almost universally used, people often sent him early versions of new modifications. An extension to the house was constructed to hold the collection—an extension that has been amply filled.
Dreihasenfenster (‘window of three hares’) in the cloister’s inner courtyard of Paderborn Cathedral
Perhaps slightly less expectedly, David was also interested in the ‘motif of the three hares’, a design in which there are three hares following each other in a circle. They each have the requisite two ears, but each shares an ear with another, so there are only three ears in total. There is some evidence that this was presented as a puzzle in ancient times, but it has also appeared in many unexpected places as a motif of some sort. It is unclear whether it was purely decorative, or intended to convey a message and carry meaning. David showed me a splendid coffee table book, which he had contributed to, about a journey along the Silk Road tracking down places where the three hares had been found. This went some way to explaining why his ever-present bag of toys had the three hares appliquéd on it.
David was full of surprises like that, but when you came to know him it was somehow less surprising. His interests were as unbounded as his youthful enthusiasm.
To paraphrase Rob Eastaway: David will be greatly missed by everyone who knew him, but his spirit will live on whenever someone casually picks up a Rubik’s cube, smiles, and starts to play with it.
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