Why must two people in London have exactly the same number of hairs on their head?
The average human head holds well under 200,000 hairs, and essentially no one has more than 500,000. London has roughly 9 million residents. Without knowing anything about any specific person, prove that at least two Londoners must have exactly the same number of head hairs.
Reveal the answer
If you tried to give every Londoner a different hair count starting from 0, you'd run out of possible counts long before you ran out of people — there are far more people than plausible hair counts, so at least two must share a count. This is the pigeonhole principle: put more items into containers than there are containers, and at least one container ends up with more than one item. Formalised by Peter Gustav Lejeune Dirichlet in 1834 as the 'box principle', it now underpins proofs across number theory and computer science.
— Peter Gustav Lejeune Dirichlet, Pigeonhole principle — 1834, 'Schubfachprinzip' (box principle)