Newsgroups: sci.math
From: hoey@zogwarg.etl.army.mil (Dan Hoey)
Date: 18 Apr 91 18:25:24 GMT
Subject: Re: Self-ref clue puzzles

david.lloyd-jo...@rose.uucp (DAVID LLOYD-JONES) writes:

>It seems to me there's a class of puzzles in which the fact that the
>puzzle is soluble constitutes one of the clues necessary for solving
>the puzzle.  Having said that, I can't for the life of my think of one.

The well-known puzzle of Monty and the three doors may be written as

  Monty Hall plays a game with the contestants on his show.  He
  shows the contestant three doors, behind one of which is a prize.
  He lets the contestant pick one of the doors, then--using his
  knowledge of where the prize is--he opens one of the unchosen
  doors that has no prize behind it.  (He can always do this because
  only one door conceals a prize).  The contestant can then choose
  whether to have what is behind the first door picked or what is
  behind the other closed door.  Which is the better choice?

for which the answer is well known; send me mail if you haven't seen
enough about this problem.  (Trying to solve it by network discussion
has been shown to yield a large number of messages, most of them wrong
in one subtle way or another.  I think it is more productive to read
about this in the rec.puzzles FAQ list.)  My point is that this
problem is usually posed in a shorter form, such as:

  Monty Hall has put a prize behind one of three doors.  He lets you
  pick one, then he opens one you didn't pick and offers to let you
  switch.  Should you?

In this second formulation we have the possibility that you are
dealing with a character I call ``Monty's Evil Twin'', who only offers
to let you switch when you picked the prize door first.  If you pick a
door without a prize, Monty's Evil Twin sends you home empty-handed.
There's also ``Saint Monty'', who doesn't offer a switch unless
switching lets you win.  It's pretty clear that you should switch
if Saint Monty lets you, but not when the Evil Twin offers.

This relates to the question of self-referential puzzles in that,
without some knowledge of the a priori distribution of Saints and Evil
Twins, you can't deduce the payoff from switching vs. staying.  Bernie
Cosell has reasoned that by assuming we are given enough information
to solve the problem, we can rule out the possibility of any but the
single-strategy Monty of the first problem statement.  (I think his
reasoning is unjustified, but Bernie says I'm being picky.  Let's call
the whole thing off.)

Back in December, Phil Servita used a statistical approach.  He noted
that if the a priori likelihood of Saint Monty is the same as that of
the Evil Twin, then your observation of being given a choice makes the
Saint twice as likely as the Twin, so they cancel out.  I would be
interested in seeing this analysis carried out on the entire space of
mixed-strategy Monties.  I think the ``default'' a priori probability
depends on how you parameterize the space, but I'm not really sure.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil