From: james@crc.ricoh.com (James Allen)
Newsgroups: rec.puzzles
Subject: Caliban's Will
Summary: Asking for a little help
Date: 26 Sep 92 09:16:34 GMT
Organization: RICOH California Research Center


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?
]]

The FAQL and I agree that the prescribed order is
	Low (first), 'Critic', Y.Y. (last)
but I'm not sure who lent Caliban an umbrella.

]]==> logic/calibans.will.s <==
]]Let T be "person who saw Caliban in a green tie."
]]Let U be "person who lent Caliban an umbrella."
]]Then the data are:
]](1) No T chooses before Low.
]](2) Either Y.Y. was in Oxford in March 1920 or the first chooser is not
]]    a U.
]](3) Either Low is second or Critic is not last.
]]
]]Consider first (3)
]]If it could be shown that Low is first, then from (3), Critic is not
]]last and therefore is second; i.e. the order is Low, Critic, Y.Y.
]]
]]Next (1)
]]If both Critic and Y.Y. were T's would require Low first and (3) then
]]gives the order Low-Critic-Y.Y., ie. (2) would be superfluous.  Hence
]]Critic and Y.Y. are not both T's.
]]
]]If neither Critic nor Y.Y. were a T, (1) would be trivially true for
]]any ordering and therefore would give no information, i.e. would be
]]superfluous.  Hence just one of Y.Y. and Critic is a T.  It follows
]]that the only possible order in which Low is not first is:
]]
]]	Not T, Low, T
]]
]]Now (2)
]]First if Y.Y was in Oxford in March 1920, nothing follows from (2) 
]]about the order and (2) is superfluous.  Hence Y.Y. was not in Oxford.
]]If Low were a U he would not, by (2) come first, and so by (1) the 
]]order would be:
]]
]]	Not T, Low, T
]]
]]i.e. (1) and (2) alone would fix an order, and (3) would be superfluous.
]]Hence Low is not a U.
]]
]]It now follows, by the arguments just given for T's under (1) that just
]]one of Y.Y. and Critic is a U.  If the same one is the T and the U (2)
]]follows from (1) (since Low is not a U); i.e (2) is superfluous.  The
]]situation is therefore:
]]	T's: just one of Y.Y. and Critic; not Low
]]	U's: the other one of Y.Y; not Low
]]It now follows that "not T, Low, T" is impossible, for the "not T" is
]]the "U" and therefore, by (2), is not first.  Hence Low is first, and
]](3) gives the order:
]]	Low, Critic, Y.Y.
]]

So far I agree with the solution.  But it seems like we don't know whether
Y.Y is a U and Critic a T or vice versa.

]]Finally, Y.Y. is a T, and Critic is a U.  For if Critic is a T, then
]]by (1) Low precedes Critic and hence (3) allows only "Low, Critic, Y.Y";
]](2) is superfluous.  I.e. Critic (only) lent Caliban an umbrella.

No. (1) and (3) also allow the ordering Y.Y. - Low - Critic.  True,
this possibility was eliminated above, but (2) was used to do so.

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.

]]
]]The problem is from _Problems Omnibus_ by Hubert Phillips,
]]Arco Publications, London, 1960.  Hubert Phillips was a noted puzzelist
]]who contributed under his own name and the pseudonyms of "Caliban",
]]"T.O. Hare", and "The Doc".
]]

Probably I'm overlooking something but I don't get it.

James