HMMT 二月 2004 · 冲刺赛 · 第 5 题

HMMT February 2004 — Guts Round — Problem 5

[6] Augustin has six 1 × 2 × π bricks. He stacks them, one on top of another, to form a tower six bricks high. Each brick can be in any orientation so long as it rests flat on top of the next brick below it (or on the floor). How many distinct heights of towers can he make? √ √

解析

Augustin has six 1 × 2 × π bricks. He stacks them, one on top of another, to form a tower six bricks high. Each brick can be in any orientation so long as it rests flat on top of the next brick below it (or on the floor). How many distinct heights of towers can he make? Solution: 28 If there are k bricks which are placed so that they contribute either 1 or 2 height, then the height of these k bricks can be any integer from k to 2 k . Furthermore, towers with different values of k cannot have the same height. Thus, for each k there are k + 1 possible tower heights, and since k is any integer from 0 to 6, there are 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 possible heights. √ √