At the beginning of the game "Clock 7 ", the arrow points to one of the seven numbers. On each turn, the arrow is rotated clockwise by the number of spaces indicated by the arrow at the beginning of the turn. For example, if "Clock 7" starts with the arrow pointing at 4 , then on the first turn, the arrow is rotated clockwise 4 spaces so that it now points at 1 . The arrow will then move 1 space on the next turn, and so on. If the arrow points at 6 after the 21 st turn, at which number did
![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-118.jpg?height=934&width=340&top_left_y=708&top_left_x=1447)
the arrow point after the first turn?
(A) 3
(B) 6
(C) 5
(D) 2
(E) 7

## Part C: Each correct answer is worth 8.
We start by making a chart in which we determine the position after the next turn given any current position:

Current Position

1

2

3

4

5

6

7

## Position after next turn

2

4

6

1

3

5

7

So for the arrow to point to the 6 after the $21^{\text {st }}$ turn, it must have pointed to the 3 after the $20^{\text {th }}$ turn (from the third row of the chart). For it to point to the 3 after the $20^{\text {th }}$ turn, it must have pointed to the 5 after the $19^{\text {th }}$ turn, and to the 6 after the $18^{\text {th }}$ turn.

This pattern now continues in a cycle, from which we can conclude the arrow pointed to the 6 after the $15^{\text {th }}$ turn, the $12^{\text {th }}$ turn, the $9^{\text {th }}$ turn, the $6^{\text {th }}$ turn and the $3^{\text {rd }}$ turn.

Since it pointed to the 6 after the $3^{\text {rd }}$ turn, it pointed to the 3 after the $2^{\text {nd }}$ turn and the 5 after the $1^{\text {st }}$ turn.

Answer: (C)