June 13, 2024


Today’s puzzle concerns a group of five power-hungry schemers who are all desperate to become the top boss. Your task will be to work out how the person of lowest status can triumph above all the others.

The puzzle is a new variant of what are often called “pirate-division” problems, and was written by Joel David Hamkins, who is currently the O’Hara Professor of Logic at the University of Notre Dame and was previously Professor of Logic at the University of Oxford.

The philosophers’ ruling council

In the ancient land of Philosophia there is a ruling council of five philosophers, linearly ranked by power and prestige, with various accompanying benefits accruing in the order of this rank. Philosopher 1 is the recognized philosopher king, the most powerful, and then philosopher 2 and so on.

It is time to pick a new council, and according to the long agreed-upon procedure, the lowest-ranked philosopher proposes a new council and ranking. The newly proposed council can in principle include any citizen at all from Philosophia—candidates are not limited to the current council members, although strangely, it usually happens that the new council is constituted by previous council members. Given the new proposal, the council votes. If a majority approve, then this is the new council and ranking; otherwise, the lowest-ranked philosopher is kicked off the council and the next lowest-ranked philosopher makes a proposal. This process continues until a new council and ranking is approved.

As mentioned above, these philosophers are a selfish crowd, all hell-bent on becoming the new council’s philosopher king. Each member prefers being on the new council above all other things, and will never vote in favor of a council that they are not on. Secondly, being on the proposed council, they would prefer to have as high a rank as possible (that is, a low number—being philosopher 1 is the best, 2 is second best, and so forth). Third, given that they will be on the council with a certain rank, they prefer that the council is as small as possible, so as not to have to share power unnecessarily (but superior rank on a larger council is preferred).

Philosopher 5, the lowest-ranked philosopher starts off, proposing a council and a ranking.

Can you suggest a proposal that guarantees philosopher 5 becomes philosopher king?

If you have never seen a “pirate-division” problem before, you may find this puzzle hard to get your head round. The solution, however, is straightforward and involves no technical knowledge.

The way to solve it is to work backwards. Think about out what happens when the council has only one philosopher, and then two philosophers, and then three, and then four and then five.

I’ll be back at 5pm UK with the solution. Please NO SPOILERS. Instead, please discuss your favourite stories about pirates.

UPDATE: Read the solution here.

Thanks to Professor Joel David Hamkins for today’s puzzle. If you are interested in the philosophy of mathematics, in particular ideas around infinity, I recommend his Substack Infinitely More, a wonderful resource that includes serialisations of his books, essays, puzzles and other writings.

I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

Football School Encyclopedia

Lastly, a quick plug for my latest book! The Football School Encyclopedia is the latest in the Football School series, aimed at 7 to 13-year-olds, which explains the world via football. It would make a rather good Christmas gift! You can find it at The Guardian Bookshop and other online retailers.



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