The relation private key --> public key is very likely unique, we can assume no collisions (I can't prove that mathematically, though), i.e. no two different private keys will give you the same public key.
To get a legacy public address there's a hashing step with RIPEMD160 that gives you a 160bit hash, thus there are mathematically obvious at max. 296 public keys with the same RIPEMD160 hash aka same legacy address.
Therefore by the initial puzzle conditions (no public key was revealed first), puzzles #161 to #256 make not much sense. This didn't occur to the puzzle creator at first but when he was nudged to it, he realized his mistake and redistributed funds from the no-sense range to the makes-sense range.
AFAIR revealing public keys for puzzles of multiples of 5 came even later (I could be wrong, don't have it crystal clear from my memory's bottom area).
Well by the same reasoning, puzzles from 81 to 159 don't make sense at all. It would have made more sense for 161-255 to only have outgoing TX, not be redistributed, because for example puzzle 254 would have somewhat similar difficulty as puzzle 127. But once #160 gets solved, then no one would be interested in solving 81 for a half-reward but double difficulty.