jmc

Question

Nine stones are arranged in a straight line. They are counted from left to right as 1,2,3, , 9, and then from right to left, so that the stone previously counted as 8 is counted as 10. The pattern is continued to the left until the stone previously counted as 1 is counted as 17. The pattern then reverses so that the stone originally counted as 2 is counted as 18, 3 as 19, and so on. The counting continues in this manner. Which of the original stones is counted as 99? Express your answer as a single digit which corresponds to the first digit assigned to that stone.

Accepted Answer

First we note that 16 stones are enumerated before the pattern repeats. Therefore, if the count enumerates a stone as

n

, then that stone is enumerated

k

for every

k \equiv n (mod 16)

(though all but the end stones are represented by two residue classes in this way).

Since

99 \equiv 3 (mod 16)

, stone number

3

is counted as 99.