[Retric]

You are treating this as a math problem not a puzzle. Is there anything written in the description that says they can all agree to an ordering that they can then apply their function to. Aka if they say box 1 maps to box 37 they need to agree what box 1 is and what box 2 is ...

[mshm]

From the page: '...Before the first prisoner enters the room, the prisoners may discuss strategy'

[Retric]

As I have pointed out in other comments discussing strategy without seeing the room makes finding a strict ordering difficult.

Picture them entering the room and the boxes are on a round filing cabinet. There is no obvious #1 to pick so some of the 100 would pick different #1's. Again, as a math problem it's reasonable to assume they can agree on an order, but it's not stated.