There is a prison, containing k prisoners. In addition to the cells of the prisoners, there is a 'special block' containing k identical cubicles arranged in a circle around the outside of a courtyard. Each cubicle contains a lightbulb, and a switch for turning the light on or off. The schedule for each day is as follows:

6am: All the lights in the special block are switched off.
7am: The prisoners are blindfolded and taken to the special block. One is placed in each cubicle and the blindfolds are removed. The prisoners may, if they wish, turn on the lights in their respective cubicles.
8am: The prisoners are blindfolded, and moved to different cubicles. Each prisoner moves one cubicle clockwise around the special block. The blindfolds are removed.
9am: The prisoners are taken back to their cells (blindfolded once more).

On day 0, one of the prisoners is singled out as 'prefect': He is told about the regime outlined above, and informed that if he can ever discover the number, k, of prisoners, then everyone will be allowed to go free. If, however, he guesses the number of prisoners and is incorrect then they will all be killed. He is allowed to spend all day thinking of a strategy, and at the end of day 0 he is allowed to send an email to all the other prisoners, outlining a suitable strategy for them to follow.

On day 1, the regime begins. The warder, having read the email containing the strategy, does his best to arrange the prisoners on each day in such a way as to hinder the prefect from discovering the number of prisoners.

In the interests of mathematical idealism, the warder orders the prisoners to refrain from attempting to send or recieve information in any way other than by turning the lights on or noticing whether they are on or not. For example, they may make no attempt to determine which cubicle they are in, or even whether they are taken to the same cubicle from day to day. If these terms are broken, all prisoners will be killed. In the interests of self-preservation, the prisoners follow this order.

Is there a strategy for the prisoners which guarantees that, whatever the warden does, the prefect can discover the number of prisoners? Or can the warden, with sufficient guile, keep them cooped up forever?

Happy thinking,

Randomer.