305 – Derived subgroups of symmetric groups

One of the problems in the last homework set is to study the derived group of the symmetric group .

Recall that if is a group and , then their commutator is defined as

.

The derived group is the subgroup of generated by the commutators.

Note that, since any permutation has the same parity as its inverse, any commutator in is even. This means that .

The following short program is Sage allows us to verify that, for , every element of is actually a commutator. The program generates a list of the commutators of , then verifies that this list is closed under products and inverses (so it is a group). It also lists the size of this group. Note that the size is precisely , so in these 4 cases:

305 – Cube moves

Here is a small catalogue of moves of the Rubik’s cube. Appropriately combining them and their natural analogues under rotations or reflections, allow us to solve Rubik’s cube starting from any (legal) position. I show the effect the moves have when applied to the solved cube.

But, first, some relevant links:

• Rubik’s Cube World  Records. Here is the video to the current record.
• Lego Rubik’s Cube Solver. And the current mechanical record, the CubeStormer II.
• Thomas Rokicki, “Twenty-two moves suffice for Rubik’s Cube®“. Math. Intelligencer, vol 32 (1), (2010), 33–40.
• Exactly 20 moves are required: “God’s number is 20“. Moves are counted here in the Half-turn metric, where any turn on any face by any angle is one move.