AMC 8 2005 Problem 24
AMC8 2005 年第 24 题
基本计数原理★★★★☆
A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?
某个计算器只有两个键［＋1］和［x2］。按其中一个键时，计算器会自动显示计算结果，例如，如果计算器最初显示“9”，当你按下［＋1］，它将显示“10”。如果你接着再按下［x2］，它将显示“20”。从显示“1”开始，你最少需要按键多少次才能得到结果“200”？
- A.8
- B.9
- C.10
- D.11
- E.12
正确答案 · Correct Answer
B
解析 · Solution
解法1
常规解法 · 倒推
从 200 往回推，偶数就 、奇数就减 1：，共 9 步。把这串操作倒过来，就是从 1 出发的按键序列 ，正好 9 次。
巧解 · 二进制读法
把 。从 1 开始，自左向右读「首位 1 之后」的每个二进制位：每读一位按一次 ，遇到数字 1 就再多按一次「+1」键。后续共 7 位（7 次 ），其中有 2 个 1（2 次「+1」），合计 7+2=9 次；这也顺带说明 8 次办不到。
Solution 1
Elementary · work backwards
Work backward from 200: halve when even, subtract 1 when odd. takes 9 steps. Reversing the operations gives the forward key sequence , so 9 keystrokes suffice.
Slick · read it in binary
Write . Starting from 1, scan the binary digits after the leading 1 from left to right: each digit costs one , and each digit equal to 1 costs an extra press of [+1]. There are 7 trailing digits (7 doublings) with two 1's among them (2 increments), giving 7+2=9 keystrokes; this also shows 8 is impossible.
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