Foundations

The sun dipped low over the bustling City of Königsberg, casting golden reflections over the Pregel River. Its waters divided the town into four distinct land masses, connected by seven foot-bridges that had become a curious point of both pride and frustration for its residents. By day, the bridges bustled with merchants and townsfolk, but as the evenings drew in, they became the source of a mysterious puzzle. Could someone, starting from any point on land, cross each of the seven bridges exactly once and return to where they began? It may sound simple, almost trivial, yet no matter how the townspeople tried, no one could find a solution.

Suppose you’ve just been hired as the new manager of Hilbert’s Infinite Hotel.
On your first day, you arrive at work only to be greeted by an infinitely long line of people in the lobby each expecting a room.
The Problem? All of the infinite number of rooms are already occupied by an infinite number of guests. The Infinite Hotel is full.