There are important things left out. Specifically, this is a math problem and putting it in terms of prisoners or humans is intended to be ignored. You should of course just know that when someone is a human in a math puzzle, what they really mean is a hypercomputer or oracle that can return correct answers for uncomputable algorithms. Here’s another example:
There is a prisoner in a cell, the same as the prisoners in the question above. Each day the warden hands him an infinitely complex algorithm, and he is released if and only if he can correctly tell the warden whether it halts or not. On what day is he released?
There are important things left out. Specifically, this is a math problem and putting it in terms of prisoners or humans is intended to be ignored. You should of course just know that when someone is a human in a math puzzle, what they really mean is a hypercomputer or oracle that can return correct answers for uncomputable algorithms. Here’s another example:
There is a prisoner in a cell, the same as the prisoners in the question above. Each day the warden hands him an infinitely complex algorithm, and he is released if and only if he can correctly tell the warden whether it halts or not. On what day is he released?
Gur svefg qnl.