One of the intriguing (or perhaps spoiling) things about this puzzle is that the solution by the famous puzzlist, Dudeny, has a flaw. You can determine the order of choosing but not who lent Caliban an umbrella.

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.”

What was the prescribed order of choosing; and who lent Caliban an umbrella?

(1) If neither Y.Y. nor ‘Critic’ had seen Caliban in a green tie then the first clause would be superfluous. Hence at least one of them has seen Caliban in a green tie. Furthermore Low is not last. The following orders are consistent with clause one.

Low, Y.Y., ‘Critic’
Low, ‘Critic’, Y.Y.
Y.Y., Low, ‘Critic’
‘Critic’, Low, Y.Y.

If both Y.Y. and ‘Critic’ have seen Caliban in a green tie then Low must be first.

(3) If ‘Critic’ is the first one to have fallen in love then Low must be second. The first and third clauses would then suffice to establish the order and clause two is superfluous; ergo ‘Critic’ was not the first to fall in love. Of the four possibilities from clause one, the order Low, Y.Y., ‘Critic’ is ruled out leaving:

Low, ‘Critic’, Y.Y.
Y.Y., Low, ‘Critic’
‘Critic’, Low, Y.Y.

(1)&(3) If both ‘Critic’ and Y.Y. had seen Caliban in a green tie then the only possible order would be Low, ‘Critic’, Y.Y. and clause two would be superfluous; ergo one and only of the two has seen Caliban in a green tie. The possibilities are:

Y.Y. has seen Caliban in a green tie:
Low, ‘Critic’, Y.Y.
‘Critic’, Low, Y.Y.

‘Critic’ has seen Caliban in a green tie:
Low, ‘Critic’, Y.Y.
Y.Y., Low, ‘Critic’

(2) If Y.Y. was in Oxford clause two is trivially true; ergo Y.Y. was not in Oxford. If nobody loaned Caliban an umbrella clause two is superfluous; ergo one or two people loaned Caliban an umbrella.

(1)&(2) If Low lent Caliban an umbrella then Low is not first. The order is then fixed because the person who saw Caliban in a green tie chooses last. Clause three would be superfluous. Ergo Low did not lend Caliban an umbrella.

(2)&(3) If both ‘Critic’ and Y.Y. lent Caliban an umbrella then Low is first; clause three then fixes the order as Low, ‘Critic’, Y.Y and clause one is superfluous. Ergo only one person (not Low) lent Caliban an umbrella.

(1)&(2)&(3)

If Y.Y. has seen Caliban in a green tie then ‘Critic’ must have lent Caliban an umbrella else there is no unambiguous solution. Likewise if ‘Critic’ has seen Caliban in a green tie then Y.Y must have lent Caliban an umbrella.

In either case the order is {Low, ‘Critic’, and Y.Y.}. We do not know who lent Caliban an umbrella but we do know that person did not see Caliban in a green tie.

In the analysis published by Dudeny (which is quite sparse) he erroneously claims that it can be deduced that ‘Critic’ lent Caliban an umbrella.