How would you move Mount Fuji?    “Adulterous Village” question

We all realize it's about recursion and can be proven with induction but we seem to block on the fact that so many nights should pass till something happens. The critical point is that for every night that passes, inaction of her peers adds information to each woman.

 

Let’s take a simpler case of three couples; all husbands are cheating on their wives:

  • Alice & Albert
  • Betty & Bob
  • Claudia & Charlie

 

The options are then: 0, 1, 2 or 3 cheating husbands.

 

“0” is not an option (the queen said there's at least one cheater).

 

Alice knows "1" is not an option either (she knows that both Bob & Charlie are cheating) but Alice thinks that both Betty & Claudia should each think the number of cheaters is "1" since if Albert is faithful then Betty thinks Charlie is the only cheater while Claudia thinks Bob is the only cheater.

 

Furthermore, Alice reasons, if Albert is faithful and Bob is thought by Betty to be faithful, Betty must assume Claudia will kill her husband on the first night (while Claudia must assume Betty will kill her husband since she must believe that Charlie is faithful).

 

As there are no shots on the first night, Alice now assumes Betty has realized Charlie is not the only cheater and that Bob cheats too. Similarly, Alice assumes that Claudia realized Bob is not the only cheater and her husband Charlie must cheat too. Clearly they will both kill their husbands on the next night.

 

No shots on the second night!

 

"Why is that?" asks herself Alice. The only option is that Betty thought there could be two cheaters without Bob being one. In other words, Alice realizes that Bob knows of Charlie and of her own Albert as cheaters.

 

Alice will shoot her husband on the third night. Since it's totally symmetrical, all wives reach the same conclusion at the same time and they will all shoot their husband on the third night.

 

Using induction, one can prove that n cheating husbands would all be shot on the nth night. The fifty cheating husbands will be all killed on the 50th night.