The Sultan’s daughters

The IMA website has the following puzzle:

A sultan has 100 daughters. A commoner may be given a chance to marry one of the daughters, but he must first pass a test. He will be presented with the daughters one at a time. As each one comes before him she will tell him the size of her dowry, and he must then decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will only allow the marriage to take place if the commoner chooses the daughter with the highest dowry. If he gets it wrong he will be executed! The commoner knows nothing about the distribution of dowries. What strategy should he adopt?

You might want to think about it first. The ‘official’ answer is …

 

 

 

 

 

One strategy the commoner could adopt is simply to pick a daughter at random. This would give him a 1/100 chance of getting the correct daughter. [But] the probability of the commoner accepting the daughter with the highest dowry is about 37% if he rejects the first 37 daughters and then chooses the next one whose dowry is greater than any he’s seen so far. This is a fraction 1/e of the total number of daughters (rounded to the nearest integer) and is significantly better than just choosing at random!

My question:

Given that the sultan knows what dowry each daughter has, in which order should he present the daughters to minimise the chance of one of them having to marry the commoner? With this in mind, what is the commoner’s best strategy? (And what has this to do with the financial crisis?)

See also

More puzzles.

Dave Marsay

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About Dave Marsay
Mathematician with an interest in 'good' reasoning.

One Response to The Sultan’s daughters

  1. yarivh says:

    I would put her right after the prettiest one 🙂
    Seriously though, I guess it would depend on what the sultan knows about the intelligence of the commoner, and how likely he is to second-guess the sultan’s strategy – but against a relatively intelligent commoner, my instinct is that having a fairly large random element to the exact positioning, but biased so there is a higher chance of placing the highest dowry relatively close to the beginning, would be a start. (If there is no random element, then any “best” strategy by the sultan could be worked out by the commoner.)

    Nice puzzle, anyway!

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