A classic problem is to push the largest possible cube through a square hole in a unit cube. That solution involves hexagons and involves the same calculation as finding the area of the largest square that can fit entirely inside a unit cube.
The late inventor of "recreational mathematics" Martin Gardner believes he was the first to take things to a higher dimension by asking for the largest cube that will fit entirely inside a unit hypercube. I know the answer but I would not attempt its derivation myself. This would be one of the more difficult puzzles ever posted on Brain Den. Even the coveted bonanova gold star seems too small a prize for anyone who finds the answer.
It's on m172.