HMMT 十一月 2024 · 冲刺赛 · 第 11 题

HMMT November 2024 — Guts Round — Problem 11

[8] A four-digit integer in base 10 is friendly if its digits are four consecutive digits in any order. A four-digit integer is shy if there exist two adjacent digits in its representation that differ by 1 . Compute the number of four-digit integers that are both friendly and shy.

解析

[8] A four-digit integer in base 10 is friendly if its digits are four consecutive digits in any order. A four-digit integer is shy if there exist two adjacent digits in its representation that differ by 1. Compute the number of four-digit integers that are both friendly and shy. Proposed by: Marin Hristov Hristov Answer: 148 Solution: There are 24 friendly numbers with digits d = a , d = a + 1, d = a + 2, d = a + 3, for 1 2 3 4 any 1 ≤ a ≤ 6, and 18 with a = 0. Out of these, only the numbers d d d d , and d d d d 2 4 1 3 3 1 4 2 are not shy, none of which has a leading digit zero. Therefore, the answer is: 6 · 24 + 18 − 7 · 2 = 148 .