In a very strange kingdom, 7 mathematicians (let's call them Ann, Ben, Cid, Dan, Eve, Flo and Guy) were sent to prison because they did a calculation which was correct, but was not in favor of the regnant king.
He was really unpredictable and he loved to gamble, so after their condemnation he declared in front of all the mathematicians: "I will select you in random order and will secretly tell each of you the number (1 to 7) of the single prison cell to which the warden will bring you immediately."
"The next morning, you will be brought back here one by one and I'll give each of you a sheet of paper and a pencil and each of you must write down a guess stating which cell each of the others was in*. Only if all you guess correctly will you be freed."
The king believed that this was a "fair" chance for them to regain freedom. "And now, I give you a few minutes to discuss a strategy."
The mathematicians knew that the chance was pretty low that all blind guesses would be correct. However, they knew something which the king was not aware of: On the way to the cells, they had to pass through a transfer room containing nothing but a clock hanging on the wall. Furthermore, they knew that each of them would be unsupervised for a short moment before being put into their respective cell.
They realized and agreed that the two hands of the clock would be the only way to communicate. A resolution of one minute would be the limit. In order to minimize any suspicion, they decided not to tilt or rotate or damage the clock, so the two hands would stay dependent and the rest of the clock would also stay intact.
The next morning, the mathematicians were brought back from their cells, again passing through this transfer room, but this time, they were constantly observed by the warden, so there was no way to touch the clock.
What could have been their strategy to regain freedom?
*thanks to @EdMurphy for the unambiguous formulation.
As @AndrewSavinykh pointed out, of course, an essential assumption is that nobody other than the mathematicians will access or change the hands of the clock. Not the king, not the warden and no forces of nature.
This puzzle I created myself and I am not aware if something similar already exists. I think I have a solution, but I'm not 100% sure. Since there are so many smart puzzle enthusiasts here, I hope to get a 100% solution or a proof that this cannot work. Any feedback is appreciated.