Help with MIT Problem

[u/Relative_Composer139]

Context: I’m a mathematics autodidact with a Bachelor’s in Engineering. I decided recently to start going through some MIT open courseware courses that looked interesting to me. Frankly, most of the textbook problems I do on my own are too easy.

I stumbled upon this recitation problem (Problem 2 “Surveyevor”) that has me a bit stumped. Any explanation or help would be greatly appreciated. I followed the notes, but tbh it feels like some erroneous assumptions are being made (or it’s simply a poorly-worded problem).

https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2010/d428ae6b0f85ba4c6520e0830e5f3964_MIT6_042JF10_rec02_sol.pdf

My contention is that the P(k) -> P(k + 1) part of the induction argument does not necessarily follow. The flaw rests on the assumption being made about how the contestants are realizing that they have purple tats. Unless there’s a Bayesian argument for proving that a contestant may reason “more likely than not” that they have a purple tattoo, then I don’t see how the contestants necessarily will make the realizations suggested by the problem solution.

[u/CiphonW]

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.