HMMT 二月 2004 · 冲刺赛 · 第 16 题

HMMT February 2004 — Guts Round — Problem 16

[8] An n -string is a string of digits formed by writing the numbers 1 , 2 , . . . , n in some order (in base ten). For example, one possible 10-string is 35728910461 What is the smallest n > 1 such that there exists a palindromic n -string?

解析

An n -string is a string of digits formed by writing the numbers 1 , 2 , . . . , n in some order (in base ten). For example, one possible 10-string is 35728910461 What is the smallest n > 1 such that there exists a palindromic n -string? Solution: 19 The following is such a string for n = 19: 9 | 18 | 7 | 16 | 5 | 14 | 3 | 12 | 1 | 10 | 11 | 2 | 13 | 4 | 15 | 6 | 17 | 8 | 19 where the vertical bars indicate breaks between the numbers. On the other hand, to see that n = 19 is the minimum, notice that only one digit can occur an odd number of times in a palindromic n -string (namely the center digit). If n ≤ 9, then (say) the digits 1 , 2 each appear once in any n -string, so we cannot have a palindrome. If 10 ≤ n ≤ 18, then 0 , 9 each appear once, and we again cannot have a palindrome. So 19 is the smallest possible n .