Exercise 526Combinatorics › Pigeonhole principleL1Added 3 Nov 2026

Consecutive integers

#pigeonhole#consecutive integers#pairs

Statement

Show that among any $n+1$ integers from $\{1,\ldots,2n\}$ , two are consecutive.

Hint

Group the integers into pairs $(1,2),(3,4),\ldots,(2n-1,2n)$ .

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Grouping. There are $n$ pairs covering $\{1,\ldots,2n\}$ .

Pigeonhole. With $n+1$ integers and $n$ pairs, some pair contains at least $2$ chosen integers.

Conclusion. Two of them are consecutive.