HMMT 二月 2001 · 冲刺赛 · 第 2 题

HMMT February 2001 — Guts Round — Problem 2

[4] Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he can satisfy any order Trevor might make. How many cubes are in the cup with the most sugar?

解析

[4] Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he can satisfy any order Trevor might make. How many cubes are in the cup with the most sugar? Solution: The only way to fill seven cups to satisfy the above condition is to use a binary scheme, so the cups must contain 1, 2, 4, 8, 16, 32, and 64 cubes of sugar.