Five balls, numbered 1 to 5 , are placed in order on a table. A sequence of steps is performed on the balls. In step 1, the rightmost ball is picked up and put in the middle of the four remaining balls. (The remaining balls are shifted to make room for the inserted ball.) Then in step 2, the leftmost ball is picked up and put in the middle of the four remaining balls. These steps repeat, with the rightmost and leftmost balls alternately picked

## Start:

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-010.jpg?height=67&width=326&top_left_y=1040&top_left_x=1322)

## After step 1:

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-010.jpg?height=63&width=323&top_left_y=1145&top_left_x=1324)

After step 2:

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-010.jpg?height=62&width=325&top_left_y=1259&top_left_x=1323)
up and put in the middle of the four remaining balls. Immediately after step $N$, the balls are in the reverse of their original order. Which of the following is a possible value of $N$ ?
(A) 2020
(B) 2028
(C) 2031
(D) 2027
(E) 2025
Starting with the balls in the order 1234 , we make a table of the positions of the balls after each of the first 10 steps:

| Step | Ball that moves | Order | after | step |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | Rightmost | 1 | 2 | 5 | 3 | 4 |
| 2 | Leftmost | 2 | 5 | 1 | 3 | 4 |
| 3 | Rightmost | 2 | 5 | 4 | 1 | 3 |
| 4 | Leftmost | 5 | 4 | 2 | 1 | 3 |
| 5 | Rightmost | 5 | 4 | 3 | 2 | 1 |
| 6 | Leftmost | 4 | 3 | 5 | 2 | 1 |
| 7 | Rightmost | 4 | 3 | 1 | 5 | 2 |
| 8 | Leftmost | 3 | 1 | 4 | 5 | 2 |
| 9 | Rightmost | 3 | 1 | 2 | 4 | 5 |
| 10 | Leftmost | 1 | 2 | 3 | 4 | 5 |

After 10 steps, the balls are in the same order as at the beginning. This means that after each successive set of 10 steps, the balls will be returned to their original order.

Since 2020 is a multiple of 10, then after 2020 steps, the balls will be in their original order. Steps 2021 through 2025 will repeat the outcomes of steps 1 through 5 above, and so after 2025 steps, the balls will be in the reverse of their original order.

Therefore, 2025 is a possible value of $N$. This argument can be adapted to check that none of 2028, 2031 and 2027 are possible values of $N$.

ANswer: (E)