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.

Richard Hamming's 1995 lecture on n-dimensional space, showing how geometric intuition about spheres and cubes breaks down in high dimensions. It is part of his course on learning to learn.

A demo of the Cubli, a 15 cm cube that uses reaction wheels to jump up onto a corner, balance there and fall in a controlled way to move. The video shows these three abilities chained into locomotion.