A king must choose among four suitors for his daughter's hand, and he wants the most astute of them.

Call a suitor 'cautious' if he will not risk death to marry the
princess.  That is, he will only guess the colour of his hat if he
believes that it is the only possible colour his hat can be.

I make the assumption that all suitors except Plato are cautious, and
that all of them (including Plato) believe that all of the other
suitors are cautious.  And they all believe that the others believe
that the others are cautious, and so on.

If Plato were cautious as well, then the strategy of all the suitors
would be as follows:

Seat 4 can only know his colour if he sees 3 black hats (probability
1/35); In this case he guesses his hat is white.

Seat 3 can only know his colour if he sees 2 black hats (probability
1/7); If seat 4 did not guess he guesses white; otherwise he guesses
black.

Seat 2 can only know his colour if he sees a black hat (probability
3/7).  If seat 3 did not guess he guesses white; otherwise he guesses
black.

Seat 1 knows his colour (probability 1).  If seat 2 did not guess he
guesses white; otherwise he guesses black.

Since Achilles chooses first, he takes seat 1.  Socrates follows with
the next best seat, seat 2.

Now Plato wants to minimize his beheading chances.  If he takes seat 3,
then if he sees two black hats he is fine.  But what if he sees two
white hats?  Then there are five remaining hats, and three of them are
black.  So he would guess black with probability 2/5 of being beheaded.
And if he sees a black and a white hat he would guess white with a
probability of 2/5 of being beheaded.

So his chance of being beheaded in seat 3 is 2/5*(1-1/7) = 12/35.
(1-1/7 is the chance of the two hats not both being black.)

But what if he takes seat 4?  Then if he sees three black hats he
guesses white and is fine.  If he sees one black and two white hats he
will not guess (50-50 chance).  If he sees two black hats and one white
(probability 12/35) he will guess white with probability 1/4 of being
beheaded, and if he sees three white hats (probability 4/35) he will
guess black with probability 1/4 of being beheaded.

So his chance of being beheaded in seat 4 is 1/4*(12/35+4/35) = 4/35.
Since he wants to minimise his chances of being beheaded, he will
choose seat 4.  This leaves Aristotle in seat 3, and the others
wondering what Plato is up to.

We know Plato lost his head.  Hence he guessed and saw either three
white hats, or two black hats and one white hat.

If Plato guesses black, then everyone realises that Plato is not a
cautious suitor, since the only possible guess by a cautious suitor in
seat 4 is white.  If he guesses white, then the others realise he is
not cautious when Aristotle does not guess.  (They know at least one of
Aristotle and Plato is not cautious.  Given Plato's unusual seat
choice, it must be him.  Occam's razor says Aristotle is not.)

What happens now depends on how rational everyone believes Plato is.
We have the following possibilities for models of Plato:

a) He would guess a colour if it were the only possibility
b) He would guess a colour provided it was better than a 50-50 chance
c) He would guess a colour provided it was at least a 50-50 chance
d) He would follow some other strategy (not rational).

Plato is of type (b); the others are of type (a), and have just
realised that Plato is not of type (a).  The question is: what type
will they assume he is?  And what type will they assume that the others
assume he is?  And so on.

If Plato were of type (c), then his chance of being beheaded in seat 4
would be 13/35.  So he would have chosen seat 3 instead.  So the others
can deduce he is not of type (c) (since he chose seat 4).

If Aristotle assumes Plato is of type (b), then he can deduce the
colour of his hat.  This did not happen, so Aristotle must have assumed
he is of type (d) (or was not prepared to take the risk).

Suppose Plato guessed black.  Then with Aristotle's not guessing the
others do not have sufficient information to make reasonable guesses.
But we know they did guess, so Plato must have guessed white.

Plato guessed white, and then Aristotle did not guess.  For this to
happen, Aristotle must have had some reason to believe that Plato was
not being cautious.  That is, Aristotle must know that Plato did not
see three black hats.  Hence Aristotle must be able to see a white hat.

Now Socrates deduces that Plato is not cautious (see above).  So he
follows the reasoning above to deduce that either his or Achilles' hat
is white.  For him to make a guess, he must see a black hat on Achilles
and deduce his hat colour is white.  Achilles goes through this
reasoning and also correctly deduces his hat colour (black).

Summary:

They sit in the following seats:

  4:  Plato       - black hat
  3:  Aristotle   - black hat
  2:  Socrates    - white hat
  1:  Achilles    - black hat

Plato sees two black hats and a white hat, and incorrectly guesses his
hat colour is white.  Aristotle sees the white hat on Socrates, and
realises that Plato is not cautious, and is not prepared to gamble on
him being rational either, so he doesn't guess.  Socrates realises
that Aristotle must have seen a white hat for his actions, and hence
his hat must be white (as Achilles is black).  Achilles realises that
for Socrates to know that his hat was white he must have seen a black
hat on Achilles, so he guesses black.

Plato is killed, and Achilles gets the girl for choosing the best seat.

-- Geoff Bailey (Fred the Wonder Worm) <ftww@cs.su.oz.au>