HMMT 二月 2000 · ORAL 赛 · 第 4 题

HMMT February 2000 — ORAL Round — Problem 4

[40] On an n × n chessboard, numbers are written on each square so that the number in a square is the average of the numbers on the adjacent squares. Show that all the numbers are the same.

解析

Let m be the smallest number, written on some square S . Then clearly all of the adjacent squares to S must have m on them (they can’t have anything smaller, since m is the smallest, and can’t have anything larger since m is the average of those adjacent numbers). Since this applies to every square on which m is written, all of the squares must have m on them. Otherwise there will be some “boundary” square with m on it, which will not have m written on one of its neighbours, a contradiction.