The problem of solving a Rubik's Cube is an instance of a more general problem called "factorization": Given a permutation group G which is generated by permutations g_1, ..., g_k, written:
G = <g_1, ..., g_k>
we seek to represent an element s in G as a short word over the generating set. That is, we want to write
s = g_a * g_b * g_c * ...
for some (a, b, c, ...) we must discover.
In Rubik's Cube speak, that's taking a scramble and finding a short sequence of moves that solves (or equally, reconstructs) that scramble.
A lot of problems can be framed as problems in finite group theory, especially in quantum physics and quantum information theory. For instance, to characterize the performance of a quantum computer, we might run a routine that executes a long sequence of so-called unitary operations from a special group called the Clifford group. But in order to run the characterization routine, we must be able to express their product of the sequence as a short word of generators—a problem identical to the Rubik's Cube solving problem.
So the Rubik's Cube gives us tremendous insight into what kinds of methods may or may not work in practice, since it's a non-trivial group that's large but tractable, abstract but realizable as a plastic toy.
That's all for exploring solutions mathematically and/or computationally. As for learning to solve a Rubik's Cube by hand? Not sure it's practical for very much, but it is pretty cool.
In Rubik's Cube speak, that's taking a scramble and finding a short sequence of moves that solves (or equally, reconstructs) that scramble.
A lot of problems can be framed as problems in finite group theory, especially in quantum physics and quantum information theory. For instance, to characterize the performance of a quantum computer, we might run a routine that executes a long sequence of so-called unitary operations from a special group called the Clifford group. But in order to run the characterization routine, we must be able to express their product of the sequence as a short word of generators—a problem identical to the Rubik's Cube solving problem.
So the Rubik's Cube gives us tremendous insight into what kinds of methods may or may not work in practice, since it's a non-trivial group that's large but tractable, abstract but realizable as a plastic toy.
That's all for exploring solutions mathematically and/or computationally. As for learning to solve a Rubik's Cube by hand? Not sure it's practical for very much, but it is pretty cool.