Research Ideas

Does anyone have any good ideas for a research paper on group theory (particularity the Rubik's Cube, but anything within the topic works)? It's for a IB math extended essay (for high school), and it seems many people here are cubers. So far i've considered investigating solvability of random configurations, but the only real way I've found is to solve it as much as possible until you either finish or get to an un-solvable state. I've also considered investigating the Devil's Algorithm (one that can solve any cube if repeated long enough) but I don't know if the rubik's cube group can be represented as a permutation group. Any ideas are very much appreciated!

Easy Math Editor

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Comments

See here and here : it does form a permutation group, as each element of the 'state-of-the-cube' group is just an arrangement (permutation) of colours of the cube. However, some colour combinations are physically impossible if you require that it be possible to complete the cube, so it doesn't form a symmetric group [containing all permutations of the set of 'colours-in-a-space']. Given that it seems the Rubik's cube has been flogged to death and beyond by people doing projects in group theory, it may be worth looking for a lesser-known group in nature.

A L - 8 years, 2 months ago

Yes, I've realized that after looking through some sample EEs. Now I'm thinking of doing an Olympiad problem and generalizing it.

Manasa Kaniselvan - 8 years, 2 months ago

Not sure if there is a solution but with the commutators for the Rubik's cube is there a minimum # of moves that any Rubik's cube configuration could be solved in? (this might already be proven / solved)

Lucian Schulte - 8 years, 2 months ago

I am trying to do an EE on Rubik's Cube, too. But my goal is to find total number of legal permutations of the cube group and find a way to mathematically describe Rubik's cube. I have represented the cube as a cyclic subgroup of a larger symmetric group, but I am having difficulties to represent algorithms for solutions. Can anyone help me? Thanks in advance

Նարեկ Գալստեան - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$