Competition problem — Combinatorics

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Combinatorics Difficulty 5.1 AIME, harder Find the answer

14. In a $3 \times 3$ grid filled with the numbers $1 \sim 9$ , the largest number in each row is colored red, and the smallest number in each row is colored green. Let $M$ be the smallest number among the three red squares, and $m$ be the largest number among the three green squares. Then $M-m$ can have $\qquad$ different values.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

14.8.

From the conditions, it is known that the values of $m$ and $M$ are positive integers from 3 to 7 (inclusive of 3 and 7).
Obviously, $M \neq m$ , so the value of $M-m$ is
$1,2,3,4,-1,-2,-3,-4 \text {. }$

Therefore, $M-m$ can have 8 different values.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.