HMMT 二月 2004 · 冲刺赛 · 第 16 题

英文解析

16. 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, tosee that n = 19 is the minimum, notice that only one digit can occur an odd numberof 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 .