We have a normal deck of 52 cards. We want to do the following magic trick:
A person from the audience chooses 5 random cards. The magician's assistant looks at the 5 cards, chooses 4 of them, hands them to the magician one by one face up and keeps the other one hidden. The magician then guesses the fifth card (the one that the assistant kept hidden) just by looking at the 4 cards he was handed in.
Is it possible to devise a strategy, so that no matter what the original 5 cards were, the trick always works?
Each card in the deck is assigned 1 to 52. Since the magician has 4 of the cards, there are only 48 possible other cards. The assistant hands the magician the cards in one of 24 permutation orders. With just the permutation, the magician can narrow down the card to one of two (48/24=2). The assistant can use his left or right hand when passing the cards to signal higher or lower of the two.