OK, let me return to the course, and try to explain why the number by itself is not enough.
I’ll include the guru, but instead of her saying “I see X blue eyes”, she’ll say “X”. That way, there’s no uncertainty about what number everyone is using, nor debate about whether a group of perfect logicians can know which of 100 numbers it’s the most logical to start with.
For whatever number X is, suppose there are X+1 people with a certain eye color. Now, if they knew, “there are at least X people with blue eyes,” they would leave on day 2 because they would see that the X people with blues eyes that they see haven’t left, and would reason that theirs must be the other set of blue eyes.
But in this case, they don’t. They know that the guru said “X” and that everyone else heard her say “X”, but they don’t have any knowledge attached to X. There is no chain of reasoning that gets them anywhere on day 2. They might notice that X happens to be the number of blue eyes that they see, but is that because there are exactly X blue eyes? There’s no inconsistency in assuming that X is just the number that came up on the dice she rolled; coincidence is insufficient to support a logical inference. They might even assume that she meant that there were at least X blue eyes, but they wouldn’t arrive at knowledge simpliciter, but knowledge assuming A. The conclusion of any syllogism would be burdened with that assumption.
Since X+1 wouldn’t work, X+2 wouldn’t either. If X is given as the number of blue eyes, the blue eyed islanders would expect the X+1 islanders they can see with blue eyes to leave on day 2. Since they don’t know anything about X, only that it is the number the guru said for no stated reason, they can’t start the logical process that gets them to their eye color. So no one else can found any other logical process on the fact that they failed to deduce their eye color: there’s nothing to be deduced from what they can see and the number X, so there’s nothing to be deduced from what they can see, the number X, and the failure of others to deduce anything from what they can see and the number X.
The same is true for any Y, X+Y: nothing can be deduced from X that will lead X+Y people to their eye color.
Make a syllogism if you disagree. Label each given with a letter, add some number as a given, and show a deductive logical process that leads to a conclusion like “my eyes are [some color]”.