A hotel with a countably infinite number of rooms is completely full, yet the manager can always find space for one more guest, a family of five, an infinitely long bus, or even an infinite fleet of buses. This deep dive unpacks Hilbert's paradox of the Grand Hotel, the thought experiment David Hilbert introduced in a 1924 lecture and George Gamow popularized in his 1947 book One Two Three Infinity. Along the way we explain why there is no last room to fall off of, why shifting every guest from room N to room N plus 1 never pushes anyone out, and why half of infinity is still infinity.

From there the episode walks through the increasingly clever room-assignment tricks mathematicians use: multiplying by two to empty every odd room, prime powers and the fundamental theorem of arithmetic, digit interleaving, triangular numbers, and binary sepa ...