>Puzzle 1:
>
>100 sages stay in a row. Each sage wears a hat, which might be either white or black. Each sage can see everyone, who stays before him, e.g. the last one sees 99 sages, 99th sees 98 and so on...
>
>They have been told, that each sage, who identifies his hat's color incorrectly, would be killed. They have a minute to find a best strategy (discuss this situation among them). They should pronounce loudly the color of his hat starting from the last one to the first one (each one hears, what the other say). Now the question: How many sages can survive and what would be their strategy?

The last sage has a 50/50 chance, but if they can get away by saying something like:

"my hat is the same color as the one in front of me and the same as the one after that," the next person can tell what color his hat is, give a similar answer and at least 98 sages get to go home for dinner. The first sage also has 50/50 chance as he has no reference to base his answer on.

Example:

100  99  98  97  96  95  94  93  92  91
  B   W   W   B   W   B   B   W   W  B

#100 could say:
My hat is the same color as #99 and the same as #98 or
My hat is different from #99 and different from #98.

Regardless of the outcome, by lookin at #98, now #99 knows his color and can say:
My hat is the same as #98 and different from #97.

Now #98 knows that his hat is different from #97 and the same as #96.

And so on.

Is this a valid solution?

Another option would be to state "my hat is black/white like/unlike the one in front of me". With this one, only #100 has 50/50 chances. The rest will know exactly what they're wearing.

I think this one's better.

Alex