Ideally, the guard would just assign a 1/7 chance of an inspection on any given day, which means it could come on any given day, but would more likely than not be within the week or at least soon thereafter; I think this is a gauss distribution for the number of days to inspection, but could be a poisson if you set the time frame differently. On a tangent, I just read that there is no theoretical limit to human life; the one year survival rates, which go down every year (as life expectancy increases with age at a rate of less than one year per year), never reach 0 percent; when this study was done, the rate plateaued at nearly 50 percent at some point. So one could calculate the odds of living to any age based on this distribution (which explains long lived people in a time of low life expectancies). The only data I could find that was recent were actuarial tables, which, stemming from a finite population and actual death rates, could not reflect this, but the patterns that do exist seem to indicate that our chance of death has a somewhat lower random element to it now. ]]>

Surprise! ]]>

But our guy can be surprised more than once. He can be surprised on every single day. The job of the jailer is to surprise him on the day he does the inspection.

Is this possible?

Surely it is in practice, but the logic says not!

]]>I think that if on a certain day you are SURE that the guard comes, and he doesn’t, then that means you’ve already been surprised.

Then, the induction goes like this:

If till sunday, the guard hadn’t come,

Therefore, he can’t come on sunday (under those conditions).

But the induction breaks down now:

So if he waits till saturday, then you’ll say “since he can’t wait till sunday, because I wouldn’t be surprised then, then he can’t come today.” But here is where the induction breaks down: if you would be sure by this induction that he would have to come on saturday, and he didn’t come, then you

Therefore, the statement can be true if there are at least two days, but would be false if there was just one.

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I suggest (can we agree?) that the problem implicitly redefines ‘surprise’ as something like ‘uncertainty’. The ‘genuine surprise’ that we’re looking for is a day on which the prisoner cannot be certain as to whether an inspection will occur.

The guard makes several truth claims: there will be an inspection, its occurrence is constrained to one of seven discrete times, and that it will be a ‘surprise’ when it occurs. And the simple meta-truth is that those claims are mutually contradictory.

Take Stephen’s two-day variation. Uncertainty requires that the inspection occur with probability 0<p1<1 day one, and p2 = 1-p1 on day two. But the condition of surprise/uncertainty precludes day 2 in advance, which in turn implies certainty for day 1, which contradicts our initial assumptions. And of course the same argument holds inductively for n>2 days.

The ‘puzzle’ is that the problem statement conceals an internal contradiction.

Rather than sorites, I’d compare it to a dollar auction. ]]>

Or take a lottery. I might be surprised that I win a grand prize, but surely it’s not surprising that *somebody* does. Or Russian Roulette: not that I’d have time for surprise, but I have no call for surprise if the gun fires.

The inspection is coming on one of the days; why not today?

That’s what I’m getting at, I suppose, by talking about the meaning of “surprise”. Sure, we know what it means. But the problem is trying to convince us that surprise is possible, in a situation in which it is not. ]]>

It’s a funny problem that to me points out a trouble with discreteness. Clearly surprise is impossible if there is only one opportunity for the guard to make an inspection. This paradox seems to be in play when there are just a few opportunities. But what if there were 1,000,000 opportunities? In fact, even as stated, with seven opportunities, the guard can surely surprise the prisoners, say by showing up on one of the first few days. What is breaking down here?

A gray area between discreteness and continuity seems not to be well understood (and is certainly outside our current mathematics) A similar example is the Sorites Paradox, the paradox of the heap.

]]>This statement presents no difficulties. So when we add the idea of “surprise”, what precisely are we adding? ]]>