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Stefan Banach: 2 puzzles worth knowing

2 puzzles from Stefan Banach, each explained in a minute and credited to the original work. Free on Savvy.

• 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 sli…

  — Hugo Steinhaus and Stefan Banach, Ham sandwich theorem

• Cut a solid ball into pieces, reassemble them, and get two identical balls

  A famous 1924 theorem says a solid ball can be cut into a finite number of pieces and reassembled, no stretching, no added material, into two solid balls, each exactly the same siz…

  — Stefan Banach and Alfred Tarski, Sur la decomposition des ensembles de points en parties respectivement congruentes

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