Level 3: Hilbert's hotel

Infinity and strange math · level 3 of 8 · rough age 9-10 (a guide for parents; anyone can start anywhere)

In level 2 you matched groups without counting them, and met the puzzle of whether there are "more" counting numbers than even numbers. In this level you check into a hotel with infinitely many rooms, where "full" does not mean "no room", and you learn why: an infinite collection can be paired with a part of itself. Then you look between numbers and find there is always another one, follow decimals that never stop, and meet Zeno's 2,500-year-old puzzle that seems to prove a runner can never catch a tortoise.

Lesson	You will be able to
A full hotel with room for one more	Find room for new guests in a full infinite hotel with a "move from room n to room n + 1" rule, and explain why it fails in an ordinary hotel
A bus of infinitely many guests	Use the "move to room 2n" rule to free all the odd rooms, and give every bus passenger a room of their own
The part can be as big as the whole	Pair the counting numbers one to one with the even numbers, the multiples of 10 and the square numbers, and explain Galileo's puzzle
Numbers between numbers	Find a fraction between any two fractions, and explain why there is no "next" fraction after 0
Decimals that never stop	Use long division to find repeating decimals like 0.333... and 0.142857142857..., and explain why they must repeat
Zeno's race	Tell Zeno's story of Achilles and the tortoise, explain why it seems to prove motion is impossible, and find when Achilles really passes

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