0

I have been thinking about a logical curiosity I came across years ago. It went something like this:

Three married women live in a circle of friends in which:

1) Any infidelity on the part of a husband is immediately known by everyone except that husband's wife. 2) Any wife who becomes certain that her husband has cheated will divorce him that day. 3) All the women are perfect reasoners 4) All the women know that 1-3 are true

Call the wives A,B, and C.

Suppose all the husbands are faithful until Superbowl weekend, during which they all cheat on their wives.

After one day, A thinks "B did not divorce her husband because she knows that C's husband cheated and C did not divorce her husband because she knows that B's husband cheated." And that situation remains unchanged for a while.

However, after several weeks, an anonymous person whom the wives all believe for some reason, sends each of them an e-mail stating "not all of your husbands have been faithful".

After the first day, A thinks "B did not divorce her husband because she knows that C's husband cheated and C did not divorce her husband because she knows that B's husband cheated. Now, however, B knows that C did not divorce her husband (and because my husband is surely innocent), B knows the reason C did not divorce her husband is because C knows that B's husband has cheated. C will reason similarly and learn that her own husband has cheated. Tomorrow, they will surely both divorce their husbands". B and C both reason exactly as A did, but from their own points of view.

After the second day, when B and C still have not divorced their husbands, A realizes that is can only mean that her husband has also been unfaithful. B and C both reason exactly the same a A did, and so, on the third day, all three of them divorce their husbands.

(Aside: I think the logic can be extended to m wives and n cheating husbands: on the nth day, all the wronged wives will divorce their husbands.)

Here is my problem:

I have not been able to figure out how any wronged wife would concluded her husband has cheated without the anonymous e-mail. Yet, in cases where there are at least 2 cheating husbands, all the wives already know that there has been cheating and they know all the other wives know there has been cheating. So, it seems that the anoymous e-mail is the key element that allows wives to reason out their husbands' guilt, in spite of the fact provides no new information to anyone.

Can someone either (A) explain how that is possible or (B) point out the flaw in my logic elsewhere (e.g., maybe there is a way to determine a husband's guilt without the letter)?

Thanks!

  • 1
    I'll look. I just figured it out anyway. Without the e-mail, A knows that B knows there has been cheating, but she doesn't know that B knows that C knows there has been cheating (and so on, for more wives). Thanks for the link though, I'll definitely check it out! – Matthew McPeak Jan 30 '17 at 04:30
  • Also check out the 'Unfaithful husbands' post that is under Related on the right. – Bram28 Jan 30 '17 at 04:36
  • I'm confused why you think anything is wrong with your ligic. Without the letter no wife could have ever concluded their husband was unfaithful. They went for weeks in ignorance. And the letter was required for exactly the logic you gave. So why do you think you need an explanation. – fleablood Jan 30 '17 at 05:20
  • @fleabag because at first it seemed to me that the letter did not contain any information that everyone did not already know. I understand now that it did. – Matthew McPeak Jan 30 '17 at 05:43
  • @RossMillikan I checked out your links. They were great and helped me understand even better. You're right, my post was a duplicate. Should I delete it somehow? – Matthew McPeak Jan 30 '17 at 13:26
  • No need to delete it. It has been marked as a duplicate. It is a good puzzle. – Ross Millikan Jan 30 '17 at 15:05

0 Answers0