There are 100 prisoners, each prisoner is given a number between 1 and 100 without knowing which number, the number can repeat and not all of the numbers must be given.

Each prisoner in his turn can guess his number, if he succeed, all of the prisoners are freed (they cant hear the other guesses).

Each prisoner is also given a list of the other prisoners numbers.

The prisoners have a chance to plot their strategy before the list is given.

What is the optimal plan?

  • 5
    $\begingroup$ Are they numbers between 1 and 100, or between 1 and 100 inclusive? Would they all be freed if just one prison gets their number right? Is it a requirement that no wrong guesses could be made? If the prisoners can't hear other guesses, can they see other prisoners? $\endgroup$ Dec 10, 2020 at 22:35
  • 1
    $\begingroup$ Number here refers to only whole numbers, correct? Like e, for instance, is between 1 and 100, but it is not a whole number. Is that allowed? $\endgroup$
    – Chipster
    Dec 11, 2020 at 0:03
  • $\begingroup$ @Chipster I don't think that's allowed. $\endgroup$ Dec 12, 2020 at 19:40

1 Answer 1


There is a familiar strategy that they can use here

They each know the sum of the other prisoners' numbers.
Assign the $n$th prisoner to guess the number which would make the total sum (of all prisoners numbers) congruent to $n$ modulo $100$.
Exactly one of them will be right.


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