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Problem 859

AMC 12 late, AIME early

Combinatorics Difficulty 4.6 Multiple choice CEMC Pascal · Canada · 2022

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 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 $N$ ?

Pick one

Official solution

Starting with the balls in the order
1 2 3 4 5, 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 $N$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.