AMC8 2005

AMC8 2005 · Q24

AMC8 2005 · Q24. It mainly tests Manipulating equations, Patterns & sequences (misc).

A certain calculator has only two keys [+1] and [×2]. 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 [×2], it would display “20.” Starting with the display “1,” what is the fewest number of keystrokes you would need to reach “200”?

某计算器只有两个按键[+1]和[×2]。按下按键时，计算器自动显示结果。例如，如果计算器原显示“9”，按[+1]则显示“10”，再按[×2]则显示“20”。从显示“1”开始，到达“200”需要的最少按键次数是多少？

(A) 8 8

(B) 9 9

(D) 11 11

(E) 12 12

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) One way to solve the problem is to work backward, either dividing by 2 if the number is even or subtracting 1 if the number is odd. $200/2 \rightarrow 100/2 \rightarrow 50/2 \rightarrow 25-1 \rightarrow 24/2 \rightarrow 12/2 \rightarrow 6/2 \rightarrow 3-1 \rightarrow 2/2 \rightarrow 1$ So if you press $[\times 2]\ [+1]\ [\times 2]\ [\times 2]\ [\times 2]\ [+1]\ [\times 2]\ [\times 2]\ [\times 2]$ or 9 keystrokes, you can reach “200” from “1.” To see that no sequence of eight keystrokes works, begin by noting that of the four possible sequences of two keystrokes, $[\times 2]\ [\times 2]$ produces the maximum result. Furthermore, $[+1]\ [\times 2]$ produces a result larger than either $[\times 2]\ [+1]$ or $[+1]\ [+1]$. So the largest possible result of a sequence of eight keystrokes is “256,” produced by either $[\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]$ or $[+1]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2].$ The second largest result is “192,” produced by $[\times 2]\ [+1]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2].$ Thus no sequence of eight keystrokes produces a result of “200.”

（B）解决这个问题的一种方法是倒推：如果数是偶数就除以 2，如果数是奇数就减 1。 $200/2 \rightarrow 100/2 \rightarrow 50/2 \rightarrow 25-1 \rightarrow 24/2 \rightarrow 12/2 \rightarrow 6/2 \rightarrow 3-1 \rightarrow 2/2 \rightarrow 1$ 因此如果你按下 $[\times 2]\ [+1]\ [\times 2]\ [\times 2]\ [\times 2]\ [+1]\ [\times 2]\ [\times 2]\ [\times 2]$（共 9 次按键），就可以从 “1” 得到 “200”。要说明不存在任何 8 次按键的序列能得到目标，先注意：在两次按键的四种可能序列中，$[\times 2]\ [\times 2]$ 产生的结果最大。此外，$[+1]\ [\times 2]$ 的结果比 $[\times 2]\ [+1]$ 或 $[+1]\ [ +1]$ 都大。所以，8 次按键序列所能得到的最大结果是 “256”，可由以下任一种产生： $[\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]$ 或 $[+1]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2].$ 第二大的结果是 “192”，由下面序列产生： $[\times 2]\ [+1]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2]\ [\times 2].$ 因此，任何 8 次按键的序列都不能得到 “200”。

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