There is a classic problem below for which Dr. Yisong Song wrote a very well written and thought out series of solutions. The variation here I've never seen anywhere but am curious. I want to work it out myself but think others may also want to give it a go.
One hundred prisoners have been newly ushered into prison. The warden tells them that starting tomorrow, each of them will be placed in an isolated cell, unable to communicate amongst each other. Each day, the warden will choose one of the prisoners uniformly at random with replacement, and place him in a central interrogation room containing only a light bulb with a toggle switch. The prisoner will be able to observe the current state of the light bulb. If he wishes, he can toggle the light bulb. He also has the option of announcing that he believes all prisoners have visited the interrogation room at some point in time. If this announcement is true, then all prisoners are set free, but if it is false, all prisoners are executed. The warden leaves, and the prisoners huddle together to discuss their fate. Can they agree on a protocol that will guarantee their freedom.
For this variation, there is a set of such 100 prisoners who are all exactly 50 years old and reasonably expect to live to 80. The light starts off and they will be taken into room on average roughly once a day. The warden may, however, send an extra prisoner of his choosing to the cell on the same day (but they still never see each other) to disrupt strategies. This excludes the use of binanary token strategy. Because the warden is not completely sadistic, a fair random number generator picks the first prisoners each day.
In their meeting, they initially agree to a simple one counter solution when one prisoner remembers (having once been a puzzle fan) that this should take about 30 years if they are lucky. He argues that each year outside of prison has an equal value but each year in prison is no better or worse than being dead. Being released on his 80th birthday is therefore worth about the same as never being released at all. He would gladly accept a 50% chance of death to be released with 20 years left of life remaining compared to a 100% chance of survival but being 79 when that chance came.
So if the value of making a guess is the number of years remaining of life multiplied by the probability of surviving the guess, what strategy should they employ to maximize this value?