AMC10 2007 B

AMC10 2007 B · Q9

AMC10 2007 B · Q9. It mainly tests Arithmetic sequences basics, Remainders & modular arithmetic.

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 “cboddg”. What letter will replace the last letter s in the message “Lee’s sis is a Mississippi miss, Chriss!”?

一种密码编码设计如下：给定消息中一个字母第一次出现时，用字母表中它右边1位的字母替换它（假设Z右边1位是A）。该字母第二次出现时，用右边1+2位替换，第三次用1+2+3位，以此类推。例如，用此编码，“banana”变成“cboddg”。消息“Lee’s sis is a Mississippi miss, Chriss!”中最后一个字母s将被什么字母替换？

(A) g g

(B) h h

(D) s s

(E) t t

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): The last s is the 12th appearance of this letter in the message, so it will be replaced by the letter that is $1+2+3+\cdots+12=\frac{1}{2}(12\cdot 13)=3\cdot 26$ letters to the right of s. Since the alphabet has 26 letters, this letter s is coded as s.

答案（D）：最后一个 s 是该字母在信息中第 12 次出现，因此它将被替换为在 s 右侧 $1+2+3+\cdots+12=\frac{1}{2}(12\cdot 13)=3\cdot 26$ 个字母的位置上的字母。由于字母表有 26 个字母，这个字母 s 被编码为 s。

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