Paradox of the Day
Hilbert's Hotel
A hotel with infinitely many rooms, all occupied, can still accommodate a new guest — by moving each guest from room N to room N+1, freeing room 1. It can even accommodate infinitely many new guests.
Why it's paradoxical
Demonstrates that infinite sets behave counterintuitively. An infinite set can be put into one-to-one correspondence with a proper subset of itself — a defining property of infinity that Georg Cantor formalized. 'Full' does not mean the same thing when applied to the infinite.