Here is the puzzle:

N hats are put on N logicians, each hat color is selected randomly: black or white.
As usual, every logician doesn’t see the hat on his own head, but sees the rest. They cannot communicate in any way possible.
Each logician at the same moment must answer the question – "what color is the hat on your head?". And there are only 3 possible answers they can say: "Black", "White" and "I don’t know".
If at least one color is named incorrectly logicians fail and die. If no one named a correct color they die just the same. Otherwise (if at least one answer is correct) – logicians survive.
As usual, they have time to discuss a strategy before the hats are put on their heads.
What’s the strategy, which gives the highest probability to survive?

It’s fairly simple to find an optimal answer for $N = 3$ ($p_{survival} = 3/4$). It’s harder, but possible to find an optimal strategy for $N = 7$ ($p_{survival} = 7/8$).
My question – is there a strategy, which has $p_{survival} > 3/4$ for $N le 6$?
How about a strategy with $p_{survival} > 7/8$ for $N = 10$?
I don’t know the answer to these questions. Please either provide such a strategy(-ies), or prove that it is impossible.
Ideally I want to know What is the maximum probability value for $N = 6$ and $N = 10$? (i.e. with a proof that we can’t do any better).

P.S. A semi-general strategy, which is optimal for $N = 3$ and $N = 7$ you can find here, but if you don’t know it, I suggest you to try to find it on your own, it’s a very fun puzzle.