This article was originally printed in Issue 426 (November/December) of the Australian Mensa magazine, TableAus.

Dr Burkard Polster, Associate Professor in the School of Mathematical Sciences, Monash University

Dr. Burkard Polster is an Associate Professor in the School of Mathematical Sciences at Monash University. His diverse research interests included finite and topological geometry, combinatorial designs, group theory and classical interpolation theory. He is also a passionate and talented communicator of mathematical ideas; His YouTube channel, Mathologer, has over one hundred thousand subscribers and he is the author of a number of popular books on the beauty of mathematics. I contacted Dr. Polster to see if he could shed light on how mathematical research helped speed-solvers crack the Rubik’s cube.

Daniel: Experienced speed-cubers are capable of ‘solving’ the Rubik’s cube in mere seconds. I’ve had the privilege of seeing this done by a world champion and it’s an incredible display of digital dexterity.
Nonetheless, I put solving in scare quotes because they are actually applying algorithms that they’ve learnt from books and the internet. As I understand it, these true solutions were developed by mathematicians and engineers, directing the tools of their trade to the problem of solving the cube.

Was it in-virtue-of the Rubik’s cube’s general popularity that it attracted the attention of mathematicians or is there something particularly interesting about this puzzle, as opposed to others?

Burkard: There are whole branches of mathematics that deal with structures that are very closely related to the Rubik’s cube (e.g. mathematical groups which are collections of symmetries that are treated like numbers). And so, in addition to the aspects to the puzzle that can be appreciated by just about anybody, a lot of mathematicians see a lot more in this puzzle.

Daniel: The standard 3x3x3 Rubik’s cube was originally marketed as having “over three billion combinations but only one solution”, but I understand that the actual number is much higher, at 43,252,003,274,489,856,000 possible combinations. Can you please try to give me a glimpse, just the intuition, behind how these techniques handle this enormous complexity?

Burkard: The sheer number of combinations actually does not tell you anything about how difficult a puzzle is. In fact, as a puzzle the Rubik’s cube is much less difficult than is generally perceived, mainly because it is possible to find algorithms that just act on very few of the cubies without disturbing the rest of the cube. What this means is that there is a lot of “room” to move pieces around within the constraints of the puzzle. You can make puzzles with a very small number of combinations that are much harder than the Rubik’s cube, for example by bandaging the Rubik’s cube in various ways.

Daniel: I read that mathematicians have studied the maximum number of moves required to solve a Rubik’s cube, sometimes called ‘God’s number’ by the cubing community. Do mathematicians know the upper bound for the number of moves required to solve the cube? Do you know how it compares to what speed-solvers achieve in competition?

Burkard: You have to be careful when you say “maximum”. What all this is about is the following: Imagine all those gazillions of configurations of the Rubik’s cube. Any one of them has a minimum number of moves that it takes to solve it. Now take the maximum number of all those numbers, that is God’s number. Here it is also important to be precise about what you mean by a move (in the quarter turn metric God’s number is 26 http://www.cube20.org/qtm/ )   I’ve heard speed cubers mention that it takes them 60-70 moves on average to solve the Cube.

Daniel: The world of cubing, it seems, owes a lot to mathematics. Have the particular mathematical problems posed by the cube, or the resulting work informed or influenced other areas of mathematics?

Burkard: Not really. You can find quite a few technical papers that have been written about the Cube and other twisty puzzles, but all of the ones I am familiar with just apply known mathematical results to derive information about these puzzles. Having said that maybe check out the following article https://www.quantamagazine.org/20140812-the-musical-magical-number-theorist/

Daniel: Are there any remaining mathematical mysteries to the Rubik’s cube or it’s variants?

Burkard: Sure, e.g. God’s number of the 4x4x4 is not known.

Daniel: I saw your impressive collection of magic cubes in one of your YouTube videos. Are these puzzles a source of inspiration for your work as a geometer? What is your favourite mathematical feature of the magic cubes? Finally, are you able to solve all of your cubes? Would you be willing to share your best time for solving the original cube?

Burkard: Yes, at least for me puzzles have always been part of what I do in life and mathematics. In many ways they reflect the way I do mathematics—I solve puzzles. Yes, I solve all my cubes, I don’t look up other peoples’ algorithms. I am not into speed solving at all. I can solve a normal 3x3x3 in about a minute using a handful of algorithms whereas Feliks & friends memorise hundreds very specialised algorithms to get to under 10 seconds.