A Small Town Affair - Solution
by
Erik Oosterwal
To get an idea of what to look for, first check out the
The Mad Hatter puzzle on
The Puzzle Page.
In order to understand what's going on in that poor little town, let's play
a game with the men in that town. If you recall, they're all perfect logicians
so this excercise should be trivial to them. Let's assume we invite
10 of the men in that town to play a game with us--we'll arange them in a
circle so that they can each see every other man, then we place a black or
white hat on their head so that they can see the hat on every head except
their own. We tell them that when the town crier stands in the center
of the circle and rings his bell, every man wearing a black hat must take
a step forward. We also tell them that at least one man will be wearing
a black hat. the town crier then proceeds to ring his bell once every
2 minutes (we have to give our perfect logicians at least a small amount
of time to sort their thoughts...)
First let's take a look at the trivial case of only one man of the 10 wearing
a black hat. When the town crier rings his bell, the perfect logician
realizes that there must be at least one black hat and he doesn't see one
on anyone else's head and concludes that his hat is black and takes a step
forward. That was simple.
The case with two men wearing black hats gets a bit more confusing. Each
of the two men wearing a black hat sees one other man wearing a black hat.
The other eight men present see two men wearing hats. The first
time the crier rings his bell, each of the two men wearing a black hat expects
the other man to step forward based on the argument given in the previous
paragraph. Of course, neither of the men steps forward because they are not
sure that they have a black hat on. As soon as they realize that the
other man must see a black hat somewhere in the crowd, and not seeing two
themselves, they realize that they must be wearing the other black hat.
When the crier rings his bell the second time, they both step forward.
If you think that was confusing to read, just imagine how confused I was
writing it.
When there are three men wearing black hats, they each see two others wearing
black hats and expect them both to step forward the second time the crier
rings his bell. Of course the other two don't, because they each see
two other black hats, then after the second bell ringing, they realize there
must be three men wearing black hats and because they only see two they conclude
that they are wearing the third black hat.
What it boils down to is that each man expects some number of men to step
forward equal to the number of black hats they see. If they see 0 black
hats, they step forward themself. If they see 1 black hat they expect
that man to step forward on the first bell ring. If he doesn't, they
conclude that they, too, are wearing a black hat. If they see 2 black
hats they expect both those men to step forward on the second bell ringing.
If they don't, they conclude that they, too, must be wearing a black
hat. This general line of reasoning goes on for as many logicians are
wearing black hats.
Now let's see how this works for the unfaithful wives in our little town...
If there had been only one unfaithful wife in town when the mayor made his
statement, each other man in town would have known about that one unfaithful
wife while the poor man whose wife had been cheating would have been totally
surprised to hear the news and taken his wife out that night to shoot her.
If there had been two unfaithful wives, each of the two husbands with the
cheating wives would have known about the other's unfaithful wife and expected
them to shoot her on that first night. When they woke up the next morning
and saw the unfaithful wife going about her business in the town, they would
have realized that there were two unfaithful wives in town, and not knowing
of any other unfaithful wives would have concluded that theirs was also
unfaithful and taken her out that night and shot her.
Again, this line of reasoning grows to fit the total number of unfaithful
wives. The 40 men whose wives were unfaithful would have known about
the other 39 and expected them to be shot on the 39th night.
When they woke up on the morning after the 39th night only
to find them going about their business in town, they would have realized
that there must have been 40 unfaithful wives and knowing only of 39 would
have concluded that theirs was also unfaithful and on the 40th
night, taken her out to the town square and shot her along side the other
39 unfaithful wives.
Copyright E. Oosterwal - 1993-2004
Back to The Puzzle Page