[DoctorOW]

Part of the point of this thought experiment is to determine what the "given number" is since there isn't a mathematically correct answer.

[lostcolony]

Sort of; 1 is just as good as 100, since whether the other army is going to attack or not is still dependent on their receiving your Nth acknowledgement, and past acknowledgements don't change the confidence of its arrival.

Which brings us to the relaxed case from the OP; make it so we only need one acknowledgement, and account for the confidence level of getting that one acknowledgement.