You guys keep trying to reduce the “perfect logician” to being no more than a calculator wherein someone has to push his buttons. In the case of the ultra android, when someone pushes his 2+2 button sequence, it might well reach out slap him as well as add the numbers.
As soon as anyone mentions a rule wherein someone must leave if they deduce their eye color, every perfect logician would instantly deduce what would be required for that to happen, whether they wanted it to or not. That means that they would instantly become aware of every logical scenario that would lead to such a deduction, not merely any one particular scenario that would depend on someone pushing their buttons. And all of them would be thinking about all possible scenarios.
One of the scenarios that would allow the deduction of their color would be dependent upon a guru saying that there was at least one blue and everyone hearing and believing that. But before the guru even had a chance to say that, they would all already know of a scenario that allowed such a deduction without the guru speaking at all, as well as any other possible scenario.
Amongst all of the possible scenarios of which a perfect logician would immediately become aware, is one wherein they all merely start counting in the fashion that I was asking about. They would all realize, before the guru said anything, that if there was anything that could allow them to all begin that counting sequence they would be able to deduce their color. So each would then immediately deduce whether there was such a common situation.
All of that would be taking place before the echo of the rule being mentioned had faded. And since even I can figure out how to deduce such a thing without the guru, it is absurd to think they such perfect logicians could not.