Three cards, one question
I take three cards--two hearts and one spade--out of a deck and show them to you. You hold the rest of the deck and close your eyes, while I place the three cards in a row face down on the table, noting which card is the spade. You then open your eyes.
Then I say to you, "You may point to one of the cards and ask me a single question which must be answerable by either yes or no. If the card you chose is a heart, I will answer the question truthfully. If it's the spade, I may answer either truthfully or falsely. After I have answered, you can point to any of the cards, including the one you initially pointed to. If you are pointing at one of the two hearts, I will give you $10. If you are pointing at the spade, you must pay me $50."
Is there a way for you to be guaranteed to win the $10?
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1 solution
Number the positions of the cards from left to right 1, 2, and 3, with 2 in the middle. Point to the card in position 2 and ask, "Is the card in position 1 a heart?"
If I answer Yes, point to the card in position 1. If the card in position 2 was a heart, then my Yes was truthful, and if the card in position 2 was a spade, then the other cards are both hearts.
If I answer No, point to the card in position 3. If the card in position 2 was a heart, then my No was truthful, and the spade is in position 1. If the card in position 2 was a spade, the other two are both hearts.