A Puzzle by Midnight Math: May

The passel of math students from the first ever Midnight Math puzzle again find themselves held captive by that rascally hyper-intelligent, pan-dimensional being.

This time they are each held in separate cells and each day, one is chosen at random to come to a special room to work on a generalized integral transform problem (each selection is completely independent of any of the previous choices).

Fun as this sounds, they can only leave once each of them has had a whack at the problem, and if any of them approach their captor before then, they will all be held forever. However, if just one of them approaches their captor once each of them has had a whack at the problem, all will go free.

The only way they can communicate is through the light in the math room: they can choose to leave it on or off. Before the they are confined to this fate, they are held in an antechamber before being ushered to their cells. In this time, they come up with a plan so that at least one of them will know when they have all banged their heads against the problem. What is a possible plan? And (optionally and a bit more tricky), given your plan, what is the expected number of days before they go free?

Send your solutions (with proof) to midnight.math@outlook.com. Correct proofs earn you a mention here, next issue.

Correct answer to last month’s puzzle:
Arash Ushani