More hat-wearing problems

[jack]

A friend on facebook linked to a variant of the 100 hats puzzle I talked about at: http://cartesiandaemon.livejournal.com/232415.html?nc=21

Suppose there are 100 scientists (or logicians, or philosophers...) and 101 hats. As in the other puzzle, the scientists are told the rules, allowed to confer, and then not allowed to speak. They're lined up all facing the same way, so the one at the back can see everyone else's hats, and the one at the front can see no-one's hats. And no-one can see their own hat.

The hats are numbered "1" to "101". The hats are placed randomly on the heads of the scientists, with one unknown hat left over. In any order the scientists can guess what hat they're wearing. But can't guess a number that was previously guessed.

What's the maximum number of scientists you can get to guess right?

If you have that, can you do it if there are 102 hats and two are left over? I genuinely don't know the answer to that one.

Hints in the comments.

Comments

[pseudomonas]

Cheat: the one at the very back has a choice of two hats (x and y). They can encode the two numbers together as 1000x + y (or whatever). This is an invalid guess, but means the others can all guess correctly assuming they work from the back forwards.

[jack]

Yes, that was my first answer :) It's not explicitly forbidden, but it's possible to do better (since if the last scientist can do equally well by naming a possibly-correct hat, they have a 50% chance of guessing right).

[gerald_duck]

I've carefully avoided reading other comments, so apologies if this has already been asked…

You don't state in this puzzle whether or not people are told the correctness of a guess. I infer from the analogy to your previous puzzle that they are not?

[jack]

It was ambiguous in the version told to me, I think. I assumed they were not told, but I'd also be interested in any answer where being told makes it easier.