The Pigeonhole Principle

Guarantees from counting

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $n$ items in $k$ boxes: some box has at least $\lceil n/k \rceil$.
• It guarantees things exist without finding them.
• Choose the “holes” cleverly: colours, months, small squares.

Steps

1. Too many pigeons
2. Socks in the dark
3. Shared birthdays
4. Three in a month
5. Hairs on heads
6. Points in a square