Who among us has not lost at least one afternoon of their life to that most seductive of toys: The Rubik’s Cube? Originally invented by the Hungarian architect Erno Rubik in 1974, this cube – although apparently not its patents – have stood the test of time.
The beauty of the Rubik’s Cube, much like the beauty of mathematics, is that it seems totally impossible at first. But as soon as you learn the solution, it becomes totally trivial. The problem is to take this jumbled up cube, and perform a series of permutations (by twisting across various axes) to get each face to display a single color. For a 3x3x3 cube there are 4.3252×1019 possible permutations to chose from. That’s quite a lot. But even so, computations taking 35-CPU years by a bank of computers at Google show that the worst possible jumbling of the cube can always be solved in 20 or fewer moves. This maximum number of moves to solve a Rubik’s cube is known as God’s Number.
So this means that for any jumbling, you’re always only 20 moves away from a solved cube. Now you see where things start to get tantalizing. Of course you may not solve the cube perfectly, that is, you might use an algorithm that ends up taking more than God’s Number. But just knowing the solution is so close at hand is already fun. The difficulty then is in coming up with an algorithm to solve the cube, and most methods do this by breaking down the algorithm in to several sets of moves, or “macros.” And these can be best thought of as operations in group theory. We can think of permutations of the cube as elements of a group, R, whose binary operation is concatenation of moves. Then building the macros to solve the cube can be thought of in terms of commutators and conjugates, see this great explainer for the full story.
So, if you are looking for a holiday gift to
occupy please your mathematical loved ones: look no further! Math’s Gear will meet all of your Rubik’s related needs with competition grade speed cubes of all dimensions. They even have the really fun looking — but I’ll admit, slightly intimidating — Skewb puzzle cube. And what better to accompany this gift of procrastination than a paper on group theory to go with it, or for the les mathematically inclined, a plain english explanation of the macros and algorithm.
Just this week there was a new record set in the 3x3x3 Rubik’s cube by Feliks Zemdegs, a 20 year old Rubik’s cube speedsolver from Australia. This maniac can solve the cube in an insane 4.73 seconds. That’s faster than you can say “Hey Feliks, can you solve this Rubik’s Cube in under 5 seconds?” And remember, the maximum number of moves to solve the cube is 20. Suffice to say, I don’t think there’s anything I can do 20 times in the span of 4.7 seconds. You can watch him break the record in the video below, and it will make you feel really happy. (h/t to Matt Parker @standupmaths for sharing these links about Zemdegs.)
Recently the German semiconductor giant Infineon built a robot that can solve a 3x3x3 Rubik’s Cube in just 0.637 seconds. That’s so fast. That’s faster than you can say “Rubik’s Cube,” it’s faster than you can say “Ru-.” Actually, if the robot threw the Rubik’s Cube really hard at your shin, in .637 seconds the sensation of pain wouldn’t even have made it to your brain yet. The video below shows it in real time and then in slow motion, and it’s pretty incredible to watch.
In you’re looking for a mathematical gift that isn’t Rubik’s related, personally I’m hoping that someone gifts me an Otrio board this year. It’s a strategy and visual perception board game that’s sort of like Set — another great mathematical game — and Tic-Tac-Toe put together. What gifts are you hoping for? Let me know @extremefriday.