HMMT 二月 2005 · TEAM1 赛 · 第 14 题

英文解析

14. [35] Suppose S tiles the natural numbers N . Show that S tiles the set { 1 , 2 , . . . , k } forsome positive integer k .
    Solution: Using the notation from above, we can find l < l such that T ∼ T . By
    1 2 l l
    1 2
    the same argument as in problem 12, as long as T 6 = N , there is a unique choice forl
    S that contains the largest integer not in T . Since the same can be said for T ,1
    l − 1 l l
    1 1 2
    we must have that T ∼ T . Continuing in this manner, we find that there mustl − 1 l − 1
    1 2
    exist some l for which N ∼ T ; then S tiles N \ T = { 1 , 2 , · · · , c − 1 } .
    l l l