Caliban's Will - Futility Closet

A curious logic problem by Cambridge mathematician Max Newman, published in Hubert Phillips’ New Statesman puzzle column in 1933:

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?

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The key is that the problem contains “no statement superfluous to its solution.” The correct solution must make use of every statement given; any solution that doesn’t do this is wrong. So, for example, Statement 1 suggests that either Y.Y. or ‘Critic’ has seen Caliban in a green tie. If this weren’t the case, then the statement would be useless and any resulting solution would be wrong. This also tells us that Low cannot choose third, because if he did then Statement 1 again would be needless. From Statement 2 we can infer that Y.Y. was not in Oxford in 1920 — if he had been, then this statement would be useless. Similarly, if no one has lent Caliban an umbrella then the second statement is not needed, so we know that someone has lent Caliban an umbrella. Very well, who lent it to him? If Low lent it, then Statement 2 informs us that Low will not choose first. Statement 1 has already told us that Low will not choose last, so if Low lent Caliban the umbrella then Low will choose second. But that can’t be — Statement 3 can be of value only if Y.Y. or ‘Critic’ chooses second. Therefore Low didn’t lend Caliban an umbrella. Is it possible that both ‘Critic’ and Y.Y. have lent Caliban an umbrella? No: In that case Statement 2 tells us that Low must choose first, and Statement 3 tells us that ‘Critic’ chooses second and Low third, so Statement 1 is not needed. So that possibility can’t be right; Caliban has received an umbrella from either ‘Critic’ or Y.Y., but not both. Similarly, if both Y.Y. and ‘Critic’ have seen Caliban in a green tie, then Statement 1 tells us that Low will choose first, but that would make Statement 2 useless. So either ‘Critic’ or Y.Y. has seen Caliban in a green tie, but not both. Okay, suppose that Y.Y. both saw Caliban in a green tie and lent him an umbrella. In that case Statement 1 would tell us that Y.Y. cannot choose first, and if that’s the case then Statement 2 is not needed. Hence if Y.Y. saw Caliban in a green tie then he can’t have lent him an umbrella, which would mean that ‘Critic’ lent Caliban an umbrella. By the same logic, if ‘Critic’ saw Caliban in a green tie, Y.Y. must have lent Caliban an umbrella. In both of those latter cases, Low must choose first (Statement 1 tells us that Low doesn’t choose last, and Statement 3 is useful only if Y.Y. or ‘Critic’ chooses second). And if that’s the case, then Statement 3 tells us that ‘Critic’ must come second and Y.Y. third. So the proper order is Low, ‘Critic’, Y.Y. (The fanciful names refer to colleagues of Phillips at New Statesman.) (Via Alex Bellos, Can You Solve My Problems?, 2017.)

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The key is that the problem contains “no statement superfluous to its solution.” The correct solution must make use of every statement given; any solution that doesn’t do this is wrong.

So, for example, Statement 1 suggests that either Y.Y. or ‘Critic’ has seen Caliban in a green tie. If this weren’t the case, then the statement would be useless and any resulting solution would be wrong. This also tells us that Low cannot choose third, because if he did then Statement 1 again would be needless.

From Statement 2 we can infer that Y.Y. was not in Oxford in 1920 — if he had been, then this statement would be useless. Similarly, if no one has lent Caliban an umbrella then the second statement is not needed, so we know that someone has lent Caliban an umbrella.

Very well, who lent it to him? If Low lent it, then Statement 2 informs us that Low will not choose first. Statement 1 has already told us that Low will not choose last, so if Low lent Caliban the umbrella then Low will choose second. But that can’t be — Statement 3 can be of value only if Y.Y. or ‘Critic’ chooses second. Therefore Low didn’t lend Caliban an umbrella.

Is it possible that both ‘Critic’ and Y.Y. have lent Caliban an umbrella? No: In that case Statement 2 tells us that Low must choose first, and Statement 3 tells us that ‘Critic’ chooses second and Low third, so Statement 1 is not needed. So that possibility can’t be right; Caliban has received an umbrella from either ‘Critic’ or Y.Y., but not both. Similarly, if both Y.Y. and ‘Critic’ have seen Caliban in a green tie, then Statement 1 tells us that Low will choose first, but that would make Statement 2 useless. So either ‘Critic’ or Y.Y. has seen Caliban in a green tie, but not both.

Okay, suppose that Y.Y. both saw Caliban in a green tie and lent him an umbrella. In that case Statement 1 would tell us that Y.Y. cannot choose first, and if that’s the case then Statement 2 is not needed. Hence if Y.Y. saw Caliban in a green tie then he can’t have lent him an umbrella, which would mean that ‘Critic’ lent Caliban an umbrella. By the same logic, if ‘Critic’ saw Caliban in a green tie, Y.Y. must have lent Caliban an umbrella.

In both of those latter cases, Low must choose first (Statement 1 tells us that Low doesn’t choose last, and Statement 3 is useful only if Y.Y. or ‘Critic’ chooses second). And if that’s the case, then Statement 3 tells us that ‘Critic’ must come second and Y.Y. third. So the proper order is Low, ‘Critic’, Y.Y.

(The fanciful names refer to colleagues of Phillips at New Statesman.)

(Via Alex Bellos, Can You Solve My Problems?, 2017.)

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Caliban’s Will