Infinity is one of those concepts that we think we understand, but most (all?) of us can't actually wrap our minds around. German mathematician David Hilbert created a thought experiment called the "Grand Hotel paradox" to demonstrate the absurd complexity of infinity.
In this thought experiment, you're responsible for managing a grand hotel with a countably infinite number of rooms, all of which have guests in them. As finite and infinite groups of new guests arrive, you need to figure out schemes to fit them into rooms without displacing the infinite number of existing guests who are already in rooms. This turns out to involve a really clever—and difficult—set of math tricks.
In this TED-Ed lesson, Jeff Dekofsky walks you through the process, and may cause your brain to explode along the way. Have fun: