douglasm wrote:I proved in the other thread that if the number of blue eyes seen is an input to the decision of how many days to skip, then success cannot be guaranteed.

(And this is why the solution spoiler of the OP stands the way it is)

However, all this means is that if I manage to find an alternative solution for this variant, it could be used by perfect logicians on the other thread and leave earlier in the other thread, and there would be a statue built after me, right?

Because, long time ago I convinced myself that it was impossible, but now I have my doubts and I always thought superrationality would enable an earlier escape. If there's an strategy that works in all cases except 1 then people should be able to escape early by not using the strategy in 3 arbitrary cases (that is, if the warden doesn't know what strategy they'll use and chooses how many people will have blue eyes, they just need to not use the strategy if they have brown eyes, if they're the person with blue eyes that wonders if their own eyes are brown, or if they're from the rest of blue eyed people that are thinking about the blue eyed person that thinks their own eyes are blue.) Or something.

The key is:

Everyone knows that nobody will leave the first day.And:

Everyone knows that everyone knows nobody will leave the first day.And:

Everyone knows that everyone knows that everyone knows that nobody will leave the first day.These three levels of knowledge are enough to be able to skip days

most of the time (in all except 1 cases as has been said)

Why three levels? Well, because you need at least 4 blue eyed people to know that nobody leaves the first day. So you need 5 (i.e. one blue eyed people that sees 4 blue eyed people, and knows they see each other, but doesn't know if they can see her) to know that everyone knows that nobody leaves the first day, and 6 to know that everyone knows that everyone knows that nobody leaves the first day.

So, it allows for this arbitrary strategy for 100 people:

-Use the decimal system.

-If you see at least 97 blue eyed people, you can pretend that day 1 is day 91

In other words, if your eyes are brown, you know that the last blue eyed person is wondering if their eyes are brown, and if the penultimate blue eyed person is wondering what they are wondering. However, the person before them IS SEEING for a fact, the last person with blue eyes. So everyone knows, that everyone knows that everyone is seeing at least 94 blue eyed people.

So, it doesn't matter if your eyes are blue or brown, you'll be having the same expectations:

The first day, if 91 people leave the island, your eyes were brown.

You were seeing 99 blue eyed people, so you know it wasn't going to happen, but some deep layer hypothetical guy didn't know it.

The second day, if 92 people leave the island, your eyes were brown.

The third day, if 93 people leave the island, your eyes were brown.

The fourth day, if 94 people leave the island, your eyes were brown.

The fifth day, if 95 people leave the island, your eyes were brown.

The sixth day, if 96 people leave the island, your eyes were brown.

The seventh day, if 97 people leave the island, your eyes were brown.

The eighth day, if 98 people leave the island, your eyes were brown.

Now things get interesting, you don't know what will happen the 9th day. All you know is that everyone was seeing at least 98 blue eyed people, so they're following the strategy of jumping to day 90.

The ninth day, if 99 people leave the island, your eyes were brown.

The tenth day 100 blue eyed people wake up and see that 99 others didn't leave, they realize that their eyes were blue and leave. The brown eyed people were seeing 100 people and waiting for day 11, they don't leave.

Does this strategy work for 99 blue eyed people?

Yes.

The first day, if 91 people leave the island, your eyes were brown.

The second day, if 92 people leave the island, your eyes were brown.

The third day, if 93 people leave the island, your eyes were brown.

The fourth day, if 94 people leave the island, your eyes were brown.

The fifth day, if 95 people leave the island, your eyes were brown.

The sixth day, if 96 people leave the island, your eyes were brown.

The seventh day, if 97 people leave the island, your eyes were brown.

The eighth day, if 98 people leave the island, your eyes were brown.

The ninth day 99 blue eyed people see each other and leave, the brown eyed people were waiting for the 10th day of the above case.

Does this strategy work for 98 blue eyed people?

Yes.

The first day, if 91 people leave the island, your eyes were brown.

The second day, if 92 people leave the island, your eyes were brown.

The third day, if 93 people leave the island, your eyes were brown.

The fourth day, if 94 people leave the island, your eyes were brown.

The fifth day, if 95 people leave the island, your eyes were brown.

The sixth day, if 96 people leave the island, your eyes were brown.

The seventh day, if 97 people leave the island, your eyes were brown.

The eighth day, 98 blue eyed people see each other and leave, the brown eyed people were waiting for the 9th day of the above case.

Does this strategy work for 97 blue eyed people?

No.

With 97 blue eyed people, each see 96, and don't use the strategy. The brown eyed people do, think they are in the above case, but are wrong.

So, clearly, we have to make the 97 leave the seventh day.

...

My idea here:

When there are 100 people on the island, some are wondering if there are 99, if there are 100, or if there are 101. The superrational beings can then agree to leave the 9th for 99, the the 10th for 100 or the 11th for 101.

In the case of 97 people, some are wondering if there are 96, 97 or 98. The superrational beings can then agree to leave the 6th for 96, the the 7th for 100 or the 8th for 101.

And so on.