Prize winners for a giveaway program are to be determined as follows: Envelopes for the first 500 entries will be numbered 1 through 500 and laid on a long table. The first official will walk along the table, opening all the envelopes. The second official will follow the first and close every even-numbered envelope. The third official will follow the second, reversing every third envelope by closing open envelopes and opening closed envelopes. The fourth official will reverse every fourth envelope, and so on until 500 officials have walked along the table and reversed envelopes. The envelopes that remain open after this procedure will be those of the contest winners. What numbers were on the winning envelopes?

Question

Prize winners for a giveaway program are to be determined as follows: Envelopes for the first 500 entries will be numbered 1 through 500 and laid on a long table. The first official will walk along the table, opening all the envelopes. The second official will follow the first and close every even-numbered envelope. The third official will follow the second, reversing every third envelope by closing open envelopes and opening closed envelopes. The fourth official will reverse every fourth envelope, and so on until 500 officials have walked along the table and reversed envelopes. The envelopes that remain open after this procedure will be those of the contest winners. What numbers were on the winning envelopes?

Answer 1

To determine the numbers on the winning envelopes, let's go step by step and track the envelope openings and closings.

First, all envelopes are open. After the first official walks along the table, all envelopes are closed because the first official closed every envelope.

Then, the second official follows and closes every even-numbered envelope. So, with every even-numbered envelope closed, the envelopes are now in the following state:

1 closed, 2 open, 3 closed, 4 open, 5 closed, 6 open, ...

Now, the third official reverses every third envelope. Starting from the first envelope, which is closed, the third official opens it. Then, moving along, the third official reverses every third envelope.

So the sequence after the third official is as follows:

1 open, 2 open, 3 open, 4 open, 5 closed, 6 open, 7 open, ...

Next, the fourth official reverses every fourth envelope. Starting from the first envelope, the fourth official closes it. Then, moving along, the fourth official reverses every fourth envelope.

The sequence after the fourth official is as follows:

1 closed, 2 open, 3 open, 4 closed, 5 closed, 6 open, 7 open, ...

This process continues with each subsequent official reversing envelopes based on their position until the 500th official.

Finally, after the 500th official, the envelopes that remain open are the winning envelopes. So, we need to find which numbers have open envelopes.

Analyzing the pattern, we realize that only the envelopes with numbers that have an odd number of divisors will remain open. Numbers with an odd number of divisors are perfect squares.

Therefore, the winning envelopes will have numbers that are perfect squares from 1 to 500.

Hence, the numbers on the winning envelopes are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, and 484.

Answer 2

Perhaps this just about identical problem will give you a clue

Read it carefully

https://www.wyzant.com/resources/answers/233973/there_are_1_000_students_and_lockers_in_a_school_the_first_student_opens_every_locker_with_an_even_locker_number_how_many_lockers_will_be_left_open