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The numbers in the drawers/envelopes are a permutation. If everyone starts with the envelope containing their number and keeps opening the envelope of the card in the previous one, they will all eventually get back to the same number via some sequence.
The sequence (or cycle) each prisoner lands may have length from 1 to 100, and there may be anything from 1 to 100 such sequences. But every prisoner is using the same set of cycles. This strategy means that the fates of the prisoners are no longer independent.
The probability that there's a cycle (there could only ever be at most one) longer than fifty turns out to be about 70%, and that corresponds to a prisoner needing to open more than fifty envelopes to get back to their number.
The solution is basically maximising the shareable information available to all the prisoners in a way which negates the "one prisoner loses, you all lose" rule by choosing a strategy which does that anyway.
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