Puzzles
Any three objects in space, however scattered, can be cut in half by one straight slice
Take any three solid shapes, scattered anywhere in space, in any sizes — say, the bread, ham and cheese of an oddly assembled sandwich. Is there always a single flat plane that slices all three exactly in half by volume, no matter how they're arranged?
Reveal the answer
Yes, always — that's the ham sandwich theorem, posed by Hugo Steinhaus and first proved by Stefan Banach in the 1930s using topology's Borsuk–Ulam theorem. It works in any number of dimensions: two shapes in a plane can always be bisected by a single straight line, and the same trick scales up to three solids and a plane, or beyond.
— Hugo Steinhaus and Stefan Banach, Ham sandwich theorem — 1938