Balanced Columns

★ 8/10combinatorics Extremal Principle Casework

Problem

The numbers $1,1,1,1,1,2,\dots,2,3,\dots,3,4,\dots,4,5,\dots,5$ (five of each) fill a $5 \times 5$ grid, one per cell, so that any two numbers in the same column differ by at most $2$ . Let $M$ be the smallest of the five column sums. Find the maximum possible $M$ .

Answer

Solution

Difficulty	8/10
Topics	combinatorics, Extremal Principle, Casework

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