A Card Trick - One puzzle a day - Puzzle Buddies

Puzzle:

A magician takes a 52 card deck and gives it to the audience. The spectators choose any 5 cards and pass these cards to the magician's assistant. The assistant announces 4 of these cards out loud. The magician responds by naming the fifth card. How does the magician "know" the fifth card?

Solution:

This is a rather lengthy solution but simple once its understood.

This is what the assistant must do. In any group of five cards, there must be at least two of the same suit. Select any 2 cards of the same suit(in case there are more than 1)and announce out loud one of them and leave the other one as the fifth card to be guessed by the magician. Thus, the magician knows the suit of the 5th card.

Next, think of the remaining three cards as Low, Medium, and High values(Ace being lowest and King being the highest). The suit is used as a tie-breaker, if necessary. We could use an alphabetical suit order: Clubs, Diamonds, Hearts, Spades. So, for example, if we have 3 of spages, 7 of clubs and 7 of hearts, then Low = 3 of spades, Medium = 7 of clubs and High = 7 of hearts.

Given Low, Medium, and High, we can encode a number between one and six as follows:

LMH = 1, LHM = 2, MLH = 3, MHL = 4, HLM = 5, HML = 6. Thus, the order in which cards 2, 3 and 4 are announced gives a number between 1 and 6 to the magician.

This still leaves us short, as the hidden card could be one of 12.

Imagine the 13 face values (Ace to King) arranged in a circle. The shorter path between two cards, counting forward from one card to the other, is never more than six places. Therefore, out of the two cards of the same suit as mentioned above, announce the card that begins the shorter path, and leave the other card as the fifth card. Now, only a number between 1 and 6 inclusive needs to be added to the first card to get to the fifth card. E.g. if we have king of hearts and 4 of hearts as the two cards, by the cyclic rule, queen + 5 = 4. So, announce the queen and then encode 4 in the last 3 cards by the order in which they are announced.

Example: The audience selects the following cards - 2C, 5D, JC, 5H, KS - and passes them tothe assistant. The only duplicate suit is clubs. Counting forward from JC to 2C is four places, so he announces JC and leaves 2C to be guessed. Of the other three cards, 5D is Low, 5H is Medium (using the suit tie-breaker), and KS is High. To represent four,the assistant uses the ordering MHL. So he announces the 3 cards in the following order 5H, KS, 5D.

To decode, the magician notes that he must count forward from JC. He notes that the natural ordering of the other three cards is: 5D, 5H, KS, and so the cards 5H, KS, 5D represent ordering MHL, which encodes the number four. He therefore counts forward four places from JC and announces the two of clubs!

Winner: None.