HMMT 二月 2005 · TEAM1 赛 · 第 14 题

HMMT February 2005 — TEAM1 Round — Problem 14

[35] Suppose S tiles the natural numbers N . Show that S tiles the set { 1 , 2 , . . . , k } for some positive integer k .

解析

[35] Suppose S tiles the natural numbers N . Show that S tiles the set { 1 , 2 , . . . , k } for some 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 for l 1 S that contains the largest integer not in T . Since the same can be said for T , l − 1 l l 1 1 2 we must have that T ∼ T . Continuing in this manner, we find that there must l − 1 l − 1 1 2 exist some l for which N ∼ T ; then S tiles N \ T = { 1 , 2 , · · · , c − 1 } . l l l