Three Three-Digit Numbers

★ 5/10combinatorics Extrema number theory Counting algebra Casework

Problem

Let $p = (a_1, a_2, \dots, a_9)$ be a permutation of $1, 2, \dots, 9$ , and define

s(p) = \overline{a_1 a_2 a_3} + \overline{a_4 a_5 a_6} + \overline{a_7 a_8 a_9},

the sum of the three three-digit numbers formed by consecutive blocks of $p$ . Let $m$ be the minimum value of $s(p)$ over all permutations $p$ for which the units digit of $s(p)$ is $0$ , and let $n$ be the number of permutations $p$ with $s(p) = m$ . Find $|m - n|$ .

Answer

Solution

Difficulty	5/10
Topics	combinatorics, Extrema, number theory, Counting, algebra, Casework

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