Although many people have given this same general idea, it seems like they have all over-complicated a very simple solution. Assuming the other members of the circle are cooperative, and pass along messages accurately:
The instructions can be passed quite openly around the circle assuming the other members are cooperative, honest and accurate.
Making an assumption that the intent is to find the number of different ways that 3 rooks could be placed on a chess board without attacking one another. In other words, not the "maximum number", which implies trying to find the right 3 pieces, but simply the number of different ways.
Brute forcing it:
The first piece has
For each of those options, it
I assume that the prisoners decide on a strategy in the presence of the warden, and that they cannot communicate after this. (So the prisoners cannot have any common secrets that the warden does not know about.) Now the warden can distribute the names over the boxes using their knowledge of the strategy. Under these assumptions the optimal strategy of the ...