Question

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!"?

$g$

$h$

$o$

$s$

$t$

Solution:

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 Problem 9

Solution:

Problem 9 in Other Years