The Monty Hall Problem
and the Banach-Tarski paradox.
The Monty Hall problem concerns the old gameshow where there are three doors: one with a prize and two with goats.
once the contestant picks one the host reveals the contents of one of the other two doors that does not contain the prize. the player is then given the choice to switch to the remaining door. Is switching beneficial?
yes it is.
The Banach-Tarski paradox uses the axiom of choice (one of the most complicated concepts in mathematics) to say that a solid ball in 3-dimensional space can be split into several (a finite number) non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball.
except that by "pieces" they mean weird noncountable measureless point scatterings.
and of course... you can't actually do that with real solids (see definition of atoms)
in fact you can make thousands of these balls through Banach-Tarski mathemagical duplication but you can't sell them (because they don't exist)
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