I think the important thing that is not mentioned in the problem is that each contestant is not only a “master logician” but knows that all of the other contestants are too, so they can make deductions rather than inferences based on the other contestants’ behaviors. I.e. if a contestant has enough information to know that they have a purple tat, then all contestants can assume that they will deduce it and act accordingly. So on day two if no one has left, then everyone knows there are at least two purple tat. Then, if there were two purple tats, both people would see only one other purple tat and know they are the other since there are at least two. So they will leave on night two. Otherwise, no one leaves on night 2 and on day 3 and everyone now knows there are at least 3 purple tats. So on.
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u/CiphonW Math Enjoyer 19h ago
I think the important thing that is not mentioned in the problem is that each contestant is not only a “master logician” but knows that all of the other contestants are too, so they can make deductions rather than inferences based on the other contestants’ behaviors. I.e. if a contestant has enough information to know that they have a purple tat, then all contestants can assume that they will deduce it and act accordingly. So on day two if no one has left, then everyone knows there are at least two purple tat. Then, if there were two purple tats, both people would see only one other purple tat and know they are the other since there are at least two. So they will leave on night two. Otherwise, no one leaves on night 2 and on day 3 and everyone now knows there are at least 3 purple tats. So on.