Prof. Tao,
Sorry for having led this web page into the discussion (since it is originally under “career advice”). However, this is what I think:
Mathemacality has a limit as applied to reality. Mathematical induction starts with K=1, but the reality is GIVEN K=100, which does not arrive as a sequence from K=1. So the sequencial reasoning isn’t valid here. The blue-eyed islanders could comprehan their reallity with our induction only when they want to (that is, if they are mathematical souls), but they don’t have too (which does not necessarily violate logic). In fact one could argue that, since induction is not necessary therefore following it is not logical, therefore there will be no effect for the islanders. Or if they are indeed mathematical souls, then I go with Philipp: They will all commit suicide even if the outsider never came. Or K= 100 is an impossible situation to be a reality (so the puzzle contradicts itself).
However, my intuition goes with the limitation of mathematical reasoning. I am reminded another puzzle, though not the same, but also about the mathematical handling of information:
(supposedly a real story, I might not get the time right) During WW2, a city (in Poland?) made an anouncement: A military exercise would be conducted some day in the following week, yet in order to get the residents of the city used to unprepared situations during the war time, the specific day of the exercise would not be given so that it would come as an suprise. A mathematician (name I forgot) immediately reasoned:
The exercise can not take place in Sunday (the last day of the week), since if so, by Saturday night, people will know (therefore not a suprise). But then it can’t be Saturday either since if so by Friday night people will know that it will happen on Saturday since the possibility of Sunday is already ruled out. But then, Friday, Thursday, Wednesday, Tuesday, Monday are not valid days either. Therefore the exercise can never happen!
The exercise still took place on Wednesday, as nobody had induced, therefore still a suprise as promised.