Friendly Hats

Been some time since I have done an actual hat puzzle. There was one recently on Tanya Khovanova’s Math Blog that I had assumed I had done before, since it was so basic. Interestingly enough, I have never actually done this one, so I might as well put it up now:

There is a room of 100 logicians wearing hats. Each hat can be either black or white, and standard hat rules apply. The logicians must simultaneously guess what colour hat they have. Every logician who guesses right is released and every one that guesses wrong is killed.

Before the hats are assigned, they logicians may get together and plan. What strategy maximizes the number of logicians you can save on average?

Thats simple enough, there are a few follow up questions that Tanya also added though:

Suppose the maximum you can save on average is N, is there a strategy that guarantees that you will save N of them?

and,

Suppose the logicians are close friends, and they do not want to see their friends die, they would rather die with them. Is there a strategy that guarantees that they will all live together or die together?

Fairly simple if you have been following all the other hat posts, most of the basic results you need to solve this out have been proven somewhere or another on this blog.