Question

一个密码编码规则如下：某个字母在给定消息中第一次出现时，替换成字母表中向右 $1$ 位的字母（规定 A 在 Z 的右边一位）。同一个字母第二次出现时，替换成向右 $1+2$ 位的字母；第三次出现时，替换成向右 $1+2+3$ 位的字母；依此类推。

例如，用这种编码，单词“banana”变为“cbodqg”。在消息“Lee's sis is a Mississippi miss, Chriss!”中，最后一个字母 s 会被替换成什么字母？

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is $1$ place to its right in the alphabet (assuming that the letter A is one place to the right of the letter Z). The second time this same letter appears in the given message, it is replaced by the letter that is $1+2$ places to the right, the third time it is replaced by the letter that is $1+2+3$ places to the right, and so on. For example, with this code the word "banana" becomes "cbodqg". What letter will replace the last letter s in the message

"Lee's sis is a Mississippi miss, Chriss!"?

$g$

$h$

$o$

$s$

$t$

答案：D

知识点：模运算三角形数

难度评级：1370

解答：

最后一个 s 是消息中字母 s 的第 $12$ 次出现，所以它向右移动 $1+2+\cdots+12=\dfrac{12\cdot 13}{2}=78$ 位。

因为 $78=3\cdot 26$ 是字母表长度 $26$ 的倍数，所以移动后回到同一个字母 s。

所以正确答案是 D。

The final s is the $12$ th appearance of the letter s in the message, so it is shifted $1+2+\cdots+12=\dfrac{12\cdot 13}{2}=78$ places to the right.

Since $78=3\cdot 26$ is a multiple of the alphabet length $26,$ the shift returns to the same letter, s.

Thus, the correct answer is D.

2007 AMC 10B 第 9 题

答案：D

难度评级：1370

解答：

其他年份的第 9 题