David Singmaster is a Professor of Mathematics at South Bank University, London. He has been a prominent figure in the cubing world since the early days. One of his publications, the Cubic Circular is available for reading on Jaap's Puzzle Page, and these can also still be purchased in paper form from David. He kindly agreed to the following interview.
Q: When did you first develop a serious interest in mathematics?
I had always had some interest in mathematics, but started university intending to be a chemist, then changed to physics. It wasn't until my last year that I took some mathematics courses in algebra and number theory, intending to do a joint major in mathematics and physics, that I got really interested. In the autumn semester, I solved a prize problem in my number theory course and won a copy of the text which I still have. In my last semester, my algebra teacher posed a question he didn't know the answer to and I solved it, eventually leading to two papers: A Maximal Generalization of Fermat's Theorem and On Polynomial Functions (mod m).
Q: When did you see your first cube and what was your reaction?
At the International Congress of Mathematicians in Helsinki in Aug 1978. John Conway, Roger Penrose and a few others had examples. I wanted one! The Hungarians were bringing them out for trade. Beryl Fletcher arranged for me to get one of the last examples that Tamas Varga had, in exchange for a copy of an Escher book.
Q: Did being a Mathematics Professor give you any extra insight on how to solve the cube?
Basically, the ideas of group theory help one to get a grip on it and provide ideas on how to proceed. More importantly, general problem solving techniques guide one's attack and these occur both in mathematics and elsewhere - I had actually taught problem solving to school teachers. The ideas of permutations, parity of permutations and the impossibility of solving the 15-14 version of the 15 puzzle were well known to me, so I didn't have any major conceptual difficulties.
Q: How long before you discovered a solution?
Knowing what the parity conservations were meant that I didn't try to find impossible processes, so it took me about two weeks to develop a general solution. Also, I had heard a bit from Conway and Penrose and I think I had assimilated the idea of conjugation from them. The last step came about 2:00 AM while lying in bed. I had seen that I could exchange two pairs of edge pieces by (F2R2)3 and I had to see that I could get any four pieces into the four positions involved, with the last being able to go in either way.
When I saw this, I knew I could achieve any even permutation and any even flip of edges. Combined with previous moves which could do the corners, I had a general solution in early Sep 1978.
However, it took some time before I could describe this in some intelligible way. Conway's notation was based on the colors of his cube and seemed very twee to me. I had a number of letters from people using x, y, z but not being clear about the handedness or which way turns went. I tried a system in the autumn which used a, b, c coordinates with definite directions and orientation, but found that anything using an abstract coordinate system was too difficult for non-mathematicians to understand. Then I realised that F, B, R, L, U, D would work very well. I don't remember this occurring in a flash of light - it evolved from combining Conway's notation with my notation. However, it now seems so obvious and natural that I continue to be amazed that those who had had a cube for months, or even years in Hungary, had not come up with the idea.
I then developed my solution method, looking for techniques that did not require major memorization. I produced a method using two basic processes: (F2R2)3 and the commutator [F,R], but it took some some fiddling. This appeared in the first version of my Notes in Feb 1979.
The next major insight was the use of double level commutators. I recall Peter McMullen telling me about the Cambridge group using moves that only affected one or two U pieces. At first this seemed silly, but then commuting this with U gives easy useful processes. I realised that the square and the cube of the commutator [F,R] gave easy moves that affected just one corner in the L face and this allowed me to produce easy processes for moving and twisting corners. From this, I built up the method given in my Notes, initially in Oct 1979, then developed in detail and made into a separate handout in Aug 1980.
Q: You were once a panelist on a puzzle game show. Could you explain what that was all about?
There are two possibilities that you might be referring to. In Nov 1981, I was on a 45 minute BBCTV show called The Adventure Game. This was a series in which three hapless members of the public are recruited to go to a mythical planet where the inhabitants enjoy playing tricks on visitors. Oddly, I've recently had communication with a fan of this series and have just transferred my video of it from Betamax to VHS and looked at it! Some of the tricks are straightforward - when you turned on the water in the kitchen, the radio came on, so you tried turning on the radio to get water. There is a mole type who tries to mislead us - she asked me to open a wine bottle and gave me a corkscrew. I looked at it and responded: "Ah, a left-handed corkscrew. I've just bought a bunch of these." and proceeded to open the wine. They cut this part out! Any way, it was good fun trying to solve problems, but it goes so fast that one doesn't even see all the clues.
In Jun 1998 to Nov 1999, I was a frequent panelist on Puzzle Panel, a BBC Radio Two half hour program. The chairman was Chris Maslanka and there would be three others selected from perhaps a dozen people that were used, mostly people with some puzzling connection who were puzzle contributors to various newspapers and magazines. We would each come with about two suitable puzzles which would be put to the panel. Chris would set a problem for the listeners and after the first program, we would get solutions and further problems sent in by listeners. Unfortunately it was never put on at a good time. We enjoyed doing the programs and got a respectable audience, but I think the BBC authorities couldn't understand what was going on!
Here's one of my problems. Find a four word sentence where all four words are pronounced the same but spelled differently.
Q: Are you married, kids, etc?
Second marriage, to Deborah in 1972. I met her while diving and doing underwater photography on an underwater archaeological dig in Sicily in 1971. The morning we got married, I got a telegram saying I'd won a research scholarship to go to Pisa for a year. One daughter, Jessica, adopted in 1976.
Q: We've had magic polyhedra for 20 years now. Apart from the inital craze of the early eighties, how has the popularity of cube type puzzles varied since that time from your perspective?
After a craze, there is a period of depression - Dudeney describes this for the 15 puzzle craze. But a puzzle like the cube was obviously destined to become a classic. However, I think the situation was definitely screwed up by Ideal. When they moved into the market, they doubled the price of the Cube, putting it into a fancy package which was totally unnecessary. Further they held up production for some time just as the craze was starting. The combination of these allowed pirates to set up and shortly after Ideal's Cubes were delivered to major outlets at a wholesale price of about #3, pirate vendors were selling cubes on the street for the same price or less. As a result, many of the major stores found themselves with stocks of unsaleable cubes and didn't reorder.
When the craze slackened, the major stores didn't reorder and cubes simply disappeared from the market for about ten years. Matchbox tried to restart an interest, but they used cheap cubes which often broke within an hour. Since the early or mid 1990s, the cube has been fairly generally available, though I'm not sure if they are as good as before. Other similar puzzles are becoming standards as well - several of Meffert's ideas are generally available.
Q: Do you see any possibility of a major return to magic puzzles by the consumer market?
I doubt we will see a major return, but they will become standards or classics and should be available in a number of forms. Just look how many 15 puzzles one sees in a typical games & puzzle shop.
Q: Just about every type of polyhedra has been made into a magic puzzle since the cube. Is there any sort of puzzle that was never made, but you wish had been?
I have a number in my collection which were only made in obscure places and small numbers - some even seem to have only been made as samples. I think the rhombic dodecahedron which I got in Poland is especially unusual. I also have a nice octahedral cube which I was sent as a sample by some oriental producer. The 4x4x4 needs a better and sturdier mechanism. I actually think that all practical polyhedra have been used, but one can adapt the standard mechanisms to have any exterior form - e.g. the Mickey Mouse and Donald Duck heads.
Q: At a rough guess, how large would your puzzle collection be and what would be your favourite puzzle of all time?
Perhaps 3000 puzzles, of which about 400 are Rubik Cubes and variants. I have a lot of illustrated cubes, and they are still appearing. Also, a number of people have made their own with strange color patterns of various sorts and have sent me examples. E.g. Kristin Wunderlich Looney's Two Dollar Cube which has the front and back of a dollar bill each covering three adjacent faces. I've also made several chimeras by mixing pieces from different shapes, e.g. a sphere and a cube, or using the cut off edge pieces from an octagonal prism to make a cube with three cut off edges which are orthogonal but non-intersecting. I also followed a suggestion of Tamas Varga and made cubes with only two or three colors.
Q: You've written many books, your solution to the cube being very popular. How many copies were sold worldwide, and is it still available from you or any other sources?
Not sure, I sold about 40,000 copies myself, but there may have been another 10000 to 20000 in the Penguin and American editions, as well as the Dutch and Spanish translations. I have supplies of my Notes on Rubik's Cube, The Handbook of Cubik Math and Rubik's Cubic Compendium. I'll attach my price list with details.
Q: What other things do you do besides teaching, puzzles, and writing? How do you relax?
Most relaxing comes from changing from one topic to another. I've taken early retirement in order to work on several books I am compiling: Sources in Recreational Mathematics; A Mathematical Gazetteer; Notes on the History of Science and Technology in London. I collect books on these fields and on cartoons, humour, language. I spend a lot of time in libraries and bookshops and I deal in books on recreational mathematics. We go to theatre, art galleries, etc. and visit friends and be tourists.
Cubic Circular Back Issues
David's price list is as follows as well as his contact details:
The CUBIC CIRCULAR started in 1981 as a newsletter for Rubik's Cube and other puzzle addicts. Eight issues eventually appeared.
- No. 1, Autumn 1981, 16 pages
- No. 2, Spring 1982, 16 pages
- No. 3/4, Spring/Summer 1982, 36 pages
- No. 5/6, Autumn/Winter 1982, 28 pages
- No. 7/8, Summer 1985, 48 pages
These 144 pages are the most comprehensive record of the Cube Craze and the incredible number of mechanical puzzles spawned by it. Everyone interested in puzzles should have a complete set.
Each year (i.e. issues 1-4 or 5-8) costs £5 (= $8) by surface mail or £6 (= $10) by air mail outside Europe. Individual issues at pro-rata prices. I am willing to accept puzzles, puzzle books, etc.
Please add 10% (minimum #1) for postage, etc. in the UK. For overseas, please double to 20% (minimum #2).
Send your name, address and payment (payable to David Singmaster) to:
87 Rodenhurst Road