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 puzzelist 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.