## Morris: The Crack that Lets the Light In

“There is a crack in everything, that’s how the light gets in” Leonard Cohen, ‘Anthem’

It is the things that don’t make sense that teach us the most.

If you want to learn something new, start with something you don’t understand, something which doesn’t make sense.

However much you know there will always be chinks in your knowledge, chinks of light that may lead you to somewhere beautiful.

Consider this:

On a Sunday afternoon your prison guard tells you that he will conduct a surprise inspection at 10am over the next seven days.

You work out that he can’t wait until next Sunday, because then it wouldn’t be a surprise.

So, could he do it on Saturday?

On which days would the inspection be a genuine surprise?

There is no consensus on this, which is why I describe it as a chink of light, a way to get at something deeper. But what do you think?

## Jonathan Lundell said,

July 2, 2010 at 10:59 pm

Is there really something deeper here? Seems to me that the only puzzle is created by a possible ambiguity in the meaning of “surprise”. If the guard tells you that he will conduct a surprise inspection tomorrow at 10am, we would simply conclude that he doesn’t understand what “surprise” means (or that he’s just being perverse).

OTOH, if all he means is that you must be prepared for an inspection on any day over the next week, but he can’t be bothered to inspect every day, then he’s accomplished what he wants.

Suppose he tells this to seven prisoners, with the intention of inspecting one cell a day, at random. The next-Sunday prisoner will have six surprises, Monday–Saturday, making up for the anticlimax on Sunday…

## Walid Damouny said,

July 3, 2010 at 4:58 am

A solution is to conduct the inspection on a time other than 10:00 a.m. and it wouldn’t matter at which day of the week.

A second solution is the conduct two inspections. The first inspection will be expected by the prisoner while the second will be a true surprise since the prisoner is not expecting it to happen.

The difficulty in pinpointing the surprise time is that the definition of “surprise” is vague. It depends on the prisoner’s expectation of the inspection timing. I don’t see a solution in picking one day of the week.

## Jonathan Lundell said,

July 4, 2010 at 9:27 pm

Another way of looking at this: suppose the guard says: an inspection will be conducted in the next seven days, at 10am on a day chosen at random.

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

## strauss said,

July 8, 2010 at 1:29 pm

Well, I think we all know what ‘surprise’ means: is there a way the guard can inspect so that at that moment the prisoners don’t know with 100% certainty that that’s the time the guard is coming? Where does the logic break down precisely?

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.

## Jonathan Lundell said,

July 8, 2010 at 6:31 pm

If that’s our definition of surprise, then it’s at least arguable that the problem is meaningless. Forget the induction issue. If I know that the guard will inspect once at 10am during the next 7 days, and he shows up on Tuesday, why should I be surprised?

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.

## Stephen Morris said,

July 8, 2010 at 6:55 pm

So what if there are only two days?

## Jonathan Lundell said,

July 9, 2010 at 3:43 pm

Does the number of days even come into it? I don’t think so (and I don’t think it’s a sorites problem).

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.

## Michael said,

July 28, 2010 at 2:14 am

I also think it has to do with the definition of surprise.

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,

and you haven’t been surprised on any day before by the fact that he hadn’t come, then you would be sure that he would come on sunday, and since this is the only day left, you wouldn’t be surprised.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

would be surprised already on saturday that he hadn’t come. Because of this, it wouldn’t matter what he did on sunday, you will already have been surprised. So on saturday, he can come, and the above definition of suprise would still hold. Since you’ve already been surprised on saturday, you will not be able to claim that you “knew he’d come on sunday”.Therefore, the statement can be true if there are at least two days, but would be false if there was just one.

## Stephen Morris said,

July 28, 2010 at 1:57 pm

I also think it’s to do with the definition of 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!

## Rebecca said,

September 12, 2010 at 4:59 pm

Having concluded that there is no day on which the visit could be a surprise, and that therefore no visit can take place, you will be surprised by any visit at all.

Surprise!

## Charlie said,

October 29, 2010 at 7:16 am

I think the way Michael looks at it comes very close to the intent such a statement is meant to convey. The intent of the prison guard (or teacher, as I originally heard this as a pop-quiz one day next week) is to make sure the prisoner/student is ever vigilant. Of course, the opposite occurs when the inspection comes early in the week and there is no chance of another inspection and therefore, no need to keep up the cell. But always doing it at the end of the week is too predictable and means the prisoner will become slack on the first few days (assuming iterations of this problem). So, much like a team that is good at passing the ball must run once in a while to keep the defense guessing, in iterations, the guard must allow some weeks when the inspection comes early to make the late week inspections effective in keeping the cell in order (we could seriously over think this in terms of game theory if we made assumptions over whether it was truly random or a day chosen by the guard or random based on a probability distribution decided on by the guard and whether there was some cost to preparing for the inspection vs a greater cost for not being prepared for it… maybe adding a reward for the guard for catching the prisoner unprepared).

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.

## Donovan said,

April 30, 2011 at 11:21 am

I was working on the same lines as Rebecca. I feel there is no such thing as a surprise once it is announced. Even if the inspection was officially canceled. Once the thought is put into the prisoner’s mind, they will not be surprised from that point forward. It will always be in the back of their mind that it will come sometime.