cubes

Mathologer shows a simple idea, the commutator, that lets anyone who can solve the first layer design their own full Rubik's cube solution. The same idea carries over to other symmetric twisty puzzles such as the Megaminx.

Numberphile with Tim Browning on the sum-of-three-cubes problem and why 33 was the lowest unsolved case. The description notes a later solution for 33.