Three princesses

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Found this one on the xkcd logic puzzle forum. I really like it.

Puzzle

You are the most eligible bachelor in the kingdom, and as such the King has invited you to his castle so that you may choose one of his three daughters to marry. The eldest princess is honest and always tells the truth. The youngest princess is dishonest and always lies. The middle princess is mischievous and tells the truth sometimes and lies the rest of the time.

As you will be forever married to one of the princesses, you want to marry the eldest (truth-teller) or the youngest (liar) because at least you know where you stand with them.

The problem is that you cannot tell which sister is which just by their appearance, and the King will only grant you ONE yes or no question which you may only address to ONE of the sisters. What yes or no question can you ask which will ensure you do not marry the middle sister?

Help

Hint

Although you might be able to solve this puzzle using a meta-question along the lines of "if I asked them what you would say if...", there is a simpler solution. Also, the question you ask can be something that all three sisters definitely know the true answer to.

Answer

Assume the three sisters are standing in a straight line. As the sister in the middle, "Is your sister on your right older than your sister on your left?" Marry the sister she indicates as younger.

Solution

There are three four possible cases. The sister standing in the middle could be the eldest, in which case she would answer truthfully. In this case, you should marry whichever sister she claims is younger, as this will be the youngest sister of the three. If the sister standing in the middle is the youngest, then she will lie to you. Thus you want to marry the sister she identifies as the younger, who will actually be the elder of the two, and as such the eldest sister of the three. Cases 3 and 4 are that you are actually asking the middle-aged sister, and she either answers truthfully or falsely. But in either case, you can select the sister she claims is younger, as that sister would either be the youngest or the eldest of the three. So always selecting the sister indicated to be the youngest leaves you with an acceptable outcome.

See also

• Three chests
• Two guards
• Angels and demons