Island wives logic puzzle

[albionmoonlight]

This is not of my creation. I have seen it a few places. Please forgive any stereotyping.

A missionary is visiting an island. There are 30 married couples on this island. The island custom is that if a woman finds out that her husband is cheating on her, she will kill him in his sleep that night. A woman will only kill her husband if she knows for a fact that he is cheating on her. Mere suspicion is not enough. The women on the island are a gossipy sort, so if a woman’s husband is cheating on her, every other woman on the island will know about it almost instantly. Put another way, every woman has knowledge of the infidelity of every husband except for her own.

Assume that every woman is a completely logical actor with great skill at deduction, and that every woman knows that every other woman on the island is a completely logical actor with great skill at deduction.

Assume further that every husband on the island is cheating on his wife.

When the missionary leaves the island, he mentions that at least one husband has been unfaithful.

Does anything happen to the men on this island? If so, what? If not, why not?

edited to correct mistake in presentation of puzzle

[Coffee Warlord]

Every single man is quite dead.

[QuikSand]

Originally posted by albionmoonlight
The island custom is that if a woman finds out that her husband is cheating on her, she will kill him as soon as she finds out.

I have heard this puzzle before (or one very much like it) but this was stated differently. As I recall from a previous telling, if a woman found out about an infidelity, she would kill the man (her husband) in his sleep that night.

If they lead to separate solutions, then perhaps we can tuck this rule away for a secondary discussion after your puzzle is resolved.

[albionmoonlight]

The odds are that I got that detail wrong in the retelling, and it is a very important detail. I will make the edit above. When I went over the puzzle in my head, I made the assumption that this condition applied without stating it.

(This, by the way, is what is known as an inadvertant hint).

[albionmoonlight]

Bump for tomorrow.

No love for the ALM puzzles . . .

[airulf]

I guess the husband did it with the missionary.

[Maple Leafs]

Originally posted by albionmoonlight
Assume that every woman is a completely logical actor with great skill at deduction...

(Cough, sputter.)

OK, I could make this assumption for pirates and even lions, but for women? You're stretching credibility.

[mckerney]

Originally posted by albionmoonlight
Assume that every woman is a completely logical actor with great skill at deduction

And how exactly are we to assume such a thing?

EDIT: Maple Leafs beat me too the point, but the question still remains.

[Maple Leafs]

OK, so I feel a little stuck on this puzzle, even though I'm fairly sure I have the answer.

Once again, we work back to smaller numbers and see if we find a pattern.

One couple: the wife realizes it must be her husband cheating, so she kills him that first night.

Two couples: each wife knows the other husband is cheating, and assumes he'll be killed that night. When he's not, each wife realizes that her husband must also be cheating, and both husbands are killed the second night.

Three couples: each wife knows the other two husbands are cheating, and assumes that the "two couples" situation above will play out and both will be killed on the second night. When they're not, each wife realizes her own husband is also cheating, and all the husbands are killed the third night.

Four couples: each wife assumes the "three couples" situation will happen, and when it doesn't they all kill their husbands on the fourth night.

Etc... so following the pattern, we find that nothing happens to the husbands until night #30, at which point they're all killed.

The only thing I don't quite understand is how the missionary triggers all this. After all, he doesn't introduce any knowledge to the island that wasn't already there. In theory, the husbands should have already started dropping before he ever arrived (unless all the cheating only started recently). Does the missionary just serve as a convenient starting point for the puzzle, or does he actually trigger the chain of events (i.e. without him spilling the beans, nothing would ever happen)?

[albionmoonlight]

First--good job on the right answer and the right logic.

Second--the missionary does trigger something. Let's use the two woman example. In that case, each woman thinks that the other woman's husband is cheating, but that her husband is not. She also thinks that the other woman believes that no one is cheating.

When the missionary says his thing, then Woman A says--"My husband is not cheating; Woman B's husband is. Because the missionary just told Woman B that someone is cheating--and because she knows that my husband is not cheating, then she will deduce that her husband is cheating and kill him tonight." Woman B uses the same logic.

You need the missionary to say his thing in order to make the non-action by the other wives mean something.

[Huckleberry]

With two couples, if they learn that at least one husband is cheating, there are two possibilities:

Only man A is cheating with woman B, only man B is cheating with woman A, or both are cheating.

In the first situation, woman A knows that she's not sleeping with man B, so she knows her husband is the one cheating and kills him the first night. Same thing for woman B in the second situation.

In the second situation with both men cheating, woman A/B will not know that her husband is cheating because she knows that the other husband is cheating and he could be the "at least one" that is cheating. But when the other woman's husband is still alive after the first night, she'll know that the other woman was thinking the same thing, so both husbands die on the second night.

So, talking it out, I find even the two couples situation to be complicated given the "at least one" statement.

Seems to me that logically every woman figures it out and every man dies. But I ain't got that kind of time.

edit - My mistake. Certainly skipped over an important part of the puzzle, that every husband is cheating.

[Maple Leafs]

Originally posted by albionmoonlight
You need the missionary to say his thing in order to make the non-action by the other wives mean something.

I can see where the missionary triggers something with two couples, since it's possible that one woman could think that nobody was cheating. But with more than two, doesn't every woman know that every other woman knows of at least one other man cheating?