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This problem is from "Caliban's Problem Book: Mathematical, Inferential
and Cryptographic Puzzles," by Hubert Phillips, S. T. Shovelton and G.
S. Marshall, T. De La Rue & Co Ltd., London, 1933. Hubert Phillips was
a noted puzzlist who wrote under his own name and the pseudonyms
of "Caliban," "Dogberry," and "Trinculo." This puzzle is the last one
in the book (#105, p. 69), and is attributed to Maxwell H. Newman
(Fielden Professor of Mathematics at Manchester University).
However, as we shall see, this puzzle is flawed. Apparently this was
never discovered by Phillips, since the puzzle was reprinted in
"Problem Omnibus: Volume I," Arco Publications, London, 1960, with
the claim that "I can say with confidence that, if there are any
errors, they are very, very, few. The credit for this goes to my
friend of many years standing, John De La Bere ... He has not only
checked the statement and solution of each problem, but has satisfied
himself that there are no redundant data and that all solutions are
unique."
Even Homer nodded.
Problem
-------
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.
1. No person who has seen me in a green tie is to choose before Low.
2. If Y.Y. was not in Oxford in 1920 the first chooser never lent me
an umbrella.
3. 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?
Solution
--------
The prescribed order of choosing is Low, 'Critic,' Y.Y. Either
'Critic' lent Caliban an umbrella and Y.Y. has seen Caliban in a green
tie or vice versa, but it is not possible to deduce which of these two
possibilities is in fact the case.
Analysis
--------
We have a set T of all people who have seen Caliban in a green tie, a
set U of all people who lent Caliban an umbrella, three distinct
ordinals C, L, and Y, and from the three statements we can deduce 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.
Intended Solution
-----------------
The argument given in the book (p. 166) is the following:
Finally, Y.Y. is a T and Critic a U.
For if Critic is a T, by (1) Low precedes Critic and hence (3) only
allows Low; Critic; Y.Y.; (2) is superfluous.
i.e. Critic (only) lent Caliban an umbrella.
However, this is a non sequitur. It is true that if Critic is a T,
then (1) implies that Low precedes Critic. But it is not the case that
then (3) implies Low; Critic; Y.Y. (3) is equally consistent with
Y.Y.; Low; Critic. It takes (2) to eliminate this ordering. Thus it
is possible that Critic is a T.
Correct Version
---------------
Probably the simplest fix to this puzzle is to drop the question about
who lent Caliban the umbrella. In the context of the story behind the
puzzle (the reading of a will), it is an irrelevant detail. It might
even be argued that the problem is more interesting because the order
of choosing can be deduced without knowing who lent the umbrella or who
saw Caliban in a green tie.
Another fix would be to ask who first fell in love, instead of who lent
Caliban the umbrella. It is possible to deduce that this was Y.Y.
However, it is also possible to change the first condition so as to
uniquely determine the umbrella lender:
1. No person who has seen me in a green tie is to choose after Critic.
Working this out is left as an exercise for the reader.