Ishari places the letters $A, B, C, D, E, F$ clockwise around a circle, starting with $A$ at the top. Jaxon writes a list beginning with $A$, and then, moving clockwise around the circle, writes down every third letter that he has not yet written. Doing this, Jaxon's list is $A, D, B, F, C, E$. Katharina mixes up the letters $L, M, N, O, P, Q$, $R, S$ and places them in the mixed-up order around a circle, starting with $L$ at the top. Jaxon writes a list beginning with $L$ and then again moving clockwise around the circle, writes down every third letter that he has not yet written. Jaxon's list is $L, M, N, O, P, Q, R, S$. Starting with $L$, what was Katharina's clockwise order?
(A) $L, O, R, N, S, Q, M, P$
(B) $L, Q, O, M, S, R, N, P$
(C) $L, R, O, M, S, Q, N, P$
(D) $L, M, N, O, P, Q, R, S$
(E) $L, O, R, M, Q, P, N, S$
Katharina placed the 8 letters in a mixed-up order around the circle.

We number the position of $L$ at the top the circle as 1, the next number moving clockwise from $L$ position 2 , and so on through to position 8 , as shown.

Jaxon begins at position 1, writes down the letter $L$, and moves clockwise writing down every third letter that he has not yet written down.

Thus, the first 3 letters that Jaxon writes down are the letters

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-029.jpg?height=385&width=358&top_left_y=1431&top_left_x=1515)

in positions 1,4 and 7 .

Continuing to move clockwise around the circle, the next three positions at which letters have not yet been written down are positions 8,2 and 3 (the letter in position 1 was already written down), and so Jaxon writes down the letter in position 3.

The next three positions at which letters have not yet been written down are 5, 6 and 8 (the letters in positions 4 and 7 were already written down), and so Jaxon writes down the letter in position 8 .

At this point, Jaxon has written the letters in positions 1, 4, 7, 3, and 8 .

The next three positions at which letters have not yet been written down are 2, 5 and 6 (the letters in positions 1, 3 and 4 were already written down), and so Jaxon writes down the letter in position 6 .

At this point, only the letters in positions 2 and 5 have not been written.

Since Jaxon left off at position 6, he skips the letter at position 2, skips the letter at position 5, and then writes down the letter at position 2 .

Finally, Jaxon writes the final letter at position 5 .

Thus in order, Jaxon writes the letters in positions $1,4,7,3,8,6,2$, and 5 .

Since Jaxon's list is $L, M, N, O, P, Q, R, S$, then $L$ is the letter in position 1 of Katharina's ordering, $M$ is the letter in position $4, N$ is the letter in position 7 , and so on.

Therefore, Katharina's clockwise order is $L, R, O, M, S, Q, N, P$.

ANSWER: (C)