# Now You Know: Rubik’s Cube turns 40

40 years ago, a Hungarian mathematician created a three-dimensional puzzle, that has
delighted and challenged people over the years who have tried to make the right moves
to solve the cube. Nov. 7 marked the 40^{th} anniversary of the Rubik’s Cube. Dr. David Plaxco, assistant professor of Mathematics,
discusses the math behind the three-dimensional puzzle, why any position on the cube
is only 20 moves away and how the Rubik’s cube can be both frustrating and fun.

**Marketing and Communications:** Dr. Plaxco, Saturday was the 40^{th} anniversary of the Rubik’s cube. Named after its founder Erno Rubik, a professor
from Budapest in Hungary, it’s widely reported that he designed the cube to help his
students understand three-dimensional problems. Can you explain the mathematics behind
the cube?

**David Plaxco:** The primary mathematics behind the Rubik’s cube is called Group Theory, which is
part of a branch of mathematics that explores the rules that dictate specific types
of structures called *groups*. A group is a set of objects and an operation on those objects that satisfies certain
rules: (1) an identity element under the operation, (2) an inverse element for each
element, (3) closure under the operation, and (4) associativity. There are several
common groups that folks deal with every day, such as integers under addition (0 is
the additive identity, a number’s opposite is its additive inverse, adding two integers
results in another integer (closure), integer addition doesn’t change if they are
re-grouped (associativity)). The group structure behind the Rubik’s cube is complicated
because it is a very large group and different from most mathematical structures that
people deal with in their day-to-day lives.

**MC:** One of the jobs (or challenges depending on who you are) is trying to solve the cube
and get each color to line up equally on each side. Apparently it took Erno over a
month to solve his first cube. What makes solving the cube so difficult?

**DP: ***Intuitively*, what makes solving the Rubik’s cube difficult is that, in order to fix something
on the cube, you have to mess something else up. So, solving the Rubik’s cube is about
messing the right things up at the right time so that you can fix them together.

*Mathematically*, what makes the cube so difficult mostly comes down to two things: the group structure
dictating moves on the cube is not commutative and is *very* large. Commutativity is a property of some mathematical structures that allows objects
to be operated together in any order without affecting the outcome. For instance,
with addition, 2 + 3 = 5 and 3 + 2 = 5, so 2 + 3 = 3 + 2. This is true for all real
numbers under addition. But** **with the Rubik’s cube, twisting layers is ** not** commutative. So, if you do a sequence of moves, the only way to undo that sequence
is to trace your steps backward in reverse order. A famous illustration of this is
called the socks and shoes principle – if you put on your socks and then put on your
shoes, when it comes time to take them off, the shoes have to come off first, then
the socks. The other thing that makes the cube difficult to solve is that there are
43,252,003,274,489,856,000 different configurations of the pieces on the cube. That
said, there are still techniques that one can learn that will allow them to solve
the puzzle in a relatively short amount of time.

**MC:** For those that can solve it, they can become speed cubers and even go to competition
for their talent. How it is possible to get so fast figuring out the cube?

**DP:** YES! Speed cubing is becoming a very popular sport*. While I am not personally a
speed cuber, I do have some insight into how folks can become very fast at the cube.
So, first, there are several different approaches to solving the cube and each approach
relies on algorithms – sequences of move (often memorized) that result in a desired
manipulation on the pieces of the cube. Speed cubers learn a *lot* of algorithms. They practice identifying specific position patterns and quickly recalling
and executing the appropriate algorithm for that situation. Speed cubers also learn
to imagine solving some (or all) of the cube mentally before actually turning a layer
(this part is pretty amazing). Finally, speed cubing requires dexterity and working
to make one’s algorithm identification and execution as fast as possible. World class
speed cubers average around 6 or 7 seconds per solve. The current world record for
a single solve is 3.47 seconds, held by Yusheng Du.

*We can debate the definition of “sport” 😝. There’s a documentary on Netflix about some of the fastest speed cubers in the world. You can watch that and decide for yourself whether speed cubing should be called a sport.

**MC:** You’ve said it’s been proven that any position is at most 20 moves away from solved.
How is that possible? And how can one determine the moves needed to solve the cube?

**DP:** This is a really strange (even to me) fact about the Rubik’s cube. Even though the
group is so large (see above), no two positions are more than 20 layer moves away
from each other. However, there is no established method for figuring out that minimal
solution! This is an example of a really interesting phenomenon in mathematics by
which mathematicians can assert a fact about the nature of solving a type of problem
without actually knowing or producing an exact solution for every scenario.

**MC:** Besides it being marketed as a fun challenge, the Rubik’s Cube is a great brain exercise.
Why do you think the Rubik’s Cube has continued to delight, and stump, people 40 years
later?

**DP:** I think there are different reasons why the cube is compelling for different people
and, of course, some people don’t quite enjoy the puzzle. Anyone who has ever tried
to solve the Rubik’s cube on their own has probably experienced it as frustrating
and enigmatic. I think that mystery does have an appeal to it – to the point that
the frustration is pleasant. In my experience, there is a threshold at which frustration
may become unpleasant. Usually, that results in a cube with the stickers peeled off.

But for those who continue at it and learn to solve the cube, I think the it provides
some comfort that a chaotic-*looking* thing can become organized and ordered... and then scrambled again! I also think
that there is an aesthetic pleasantness to the Rubik’s cube that can convey a sense
of connectedness between the pieces, even as they are moving. It’s also not just *one* thing. It’s a fidget toy and a puzzle, yes, but it’s also a machine and a sculpture
and a *massive* mathematical space to be explored.

**MC:** Lastly, can you share some of your best tips to solve the Rubik’s cube?

**DP:** Most people (including me) don’t figure out how to solve the cube on their own. There
are several great free tutorials on the internet. If you want to learn on your own,
then I would recommend buying a stickerless budget speed cube (Google that phrase,
you can find them for under $5). It’s relatively easy to learn how to take these apart
and re-assemble them as solved (you can remove a screw hidden under the center of
a face). If you’re trying to teach yourself, it’s really helpful to be able to “reset”
the cube. Bonus: re-assembling a cube is kind of like solving a jig-saw puzzle, so
you get two puzzles in one!