From: chris@questrel.com (Chris Cole)
Subject: Re: Caliban's Will
Organization: Questrel, Inc.
Date: Sun, 27 Sep 1992 00:58:07 GMT

In article <1992Sep26.091634.3029@crc.ricoh.com> james@crc.ricoh.com (James Allen) writes:
>
>Can someone help me with Caliban's Will?  There's one step in the
>solution I don't understand.
>
>In the rec.puzzles FAQL:
>]]==> logic/calibans.will.p <==
>]]	----------------------------------------------
>]]	|       Caliban's Will by M.H. Newman        |
>]]	----------------------------------------------
>]]
>]]When Caliban's will was opened it was found to contain the following
>]]clause:
>]]
>]]"I leave ten of my books to each of Low, Y.Y., and 'Critic,' who are
>]]to choose in a certain order.
>]]
>]]No person who has seen me in a green tie is to choose before Low.
>]]
>]]If Y.Y. was not in Oxford in 1920 the first chooser never lent me
>]]an umbrella.
>]]
>]]If Y.Y. or 'Critic' has second choice, 'Critic' comes before the one
>]]who first fell in love."
>]]
>]]Unfortunately Low, Y.Y., and 'Critic' could not remember any of the
>]]relevant facts; but the family solicitor pointed out that, assuming the
>]]problem to be properly constructed (i.e. assuming it to contain no
>]]statement superfluous to its solution) the relevant data and order
>]]could be inferred.
>]]
>]]What was the prescribed order of choosing; and who lent Caliban an
>]]umbrella?
>
>Clause (3) in the will may seem vacuous when Low chooses second, but in
>fact, when Y.Y. was "first to fall in love" and the only one to lend
>Caliban an umbrella and Critic the only one to see Caliban in a green tie,
>all three of clauses (1),(2),(3) are necessary to dictate an ordering:
>only (1) eliminates Critic first; only (2) eliminates Y.Y-Low-Critic,
>and only (3) eliminates Low-Y.Y.-Critic.
>
>Probably I'm overlooking something but I don't get it.
>

Well, let's see.  We have three distinct ordinals C, L, and Y and we
know three facts:

(1) for all x in T, x>L
(2) for all x in U, x>1
(3) L=2 or C<3

In addition, we know that none of these three facts is useless;
in particular, no fact implies another.

There are a lot of cases, so let's see if we can eliminate some.

If T is empty, (1) is useless, so T is not empty.
If U is empty, (2) is useless, so U is not empty.
L is not in T from (1).  (This is debatable, but does not change the answer.)
If all x in T are also in U, then (1) implies (2).  So some x in U are not in T.
If L is in U, then (2) implies (3) (this is cute).  So L is not in U.

OK, that's much better.  Now we have only the cases:
(a) T = {C}, U = {Y}
(b) T = {Y}, U = {C}

Let's translate the equations for the two cases:

        Case (a)    Case (b)
(1)     C>L         Y>L
(2)     Y>1         C>1
(3)     L=2 or C<3  L=2 or C<3

If we look at all possible orders, we find that they are eliminated as follows:

        Case (a)    Case (b)
1 2 3
C L Y   (1)         (2)
C Y L   (1)         (2)
L C Y
L Y C   (3)         (3)
Y C L   (1) (2)     (2) (1)
Y L C   (2)         (1)

So, in both cases, the only possible order is L C Y.  But there are two
possible cases, and there is not enough information to determine which is
which.  Actually, this is evident from the following argument:

Let S(C,Y) be the set of orders that are eliminated by rules (1) and
(2).  Rule (3) only eliminates one possible order, and this order
cannot be in S(C,Y) or (3) would be useless.  But rules (1) and (2) are
symmetric between C and Y, so then S(Y,C) must equal S(C,Y).  Therefore,
any assignment of C and Y to T and U must have an equally valid assignment
with Y and C switched.

In conclusion, I believe Mr. Allen is correct and I will regretfully remove
this problem from the FAQL as defective.
Newsgroups: rec.puzzles