Albert, Ben, Cyril, Dan, Erik, Filip, and Gábo entered a cave with many doors one after another in this order. They only change their order when passing through a door, and they do so as follows: the first in line opens the door, holds it for everyone else, and then takes the last position. So, the first door is opened by Albert, the second by Ben, and so on.

Who will open the 100th door? 

(K. Pazourek)
The order of the cavers changes as follows:

input: A, B, C, D, E, F, G

1. door: B, C, D, E, F, G, A
2. door: C, D, E, F, G, A, B
3. door: A, B, C, D, E, F, G
4. door: B, C, D, E, F, G, A
5. door: C, D, E, F, G, A, B
6. door: A, B, C, D, E, F, G
7. door: B, C, D, E, F, G, A
8. door: C, D, E, F, G, A, B

In every seventh door, their order is the same.

The nearest multiple of seven less than 100 is $98 (7 \cdot 14 = 98)$. Thus, the 98th door (as well as the 7th, 14th, etc.) is passed through by the cavers in their original order. The next, 99th door (as well as the 1st, 8th, 15th, etc.) is opened by Adam. The next, 100th door (as well as the 2nd, 9th, 16th, etc.) is opened by Ben.

Evaluation. 2 points for observing that the order of the cavers repeats with a period corresponding to their number; 2 points for revealing the original order in the 98th door; 2 points for solving the problem.