Decoding Cards

You and your partner are presented with a challenge using a standard deck of cards. Five cards will be drawn randomly from the deck. Your task is to devise a system where you can hand over four of these five cards to your partner. Using only the information conveyed through these four cards, your partner must be able to deduce the identity of the fifth (hidden) card. Is there a method that guarantees success every time? If so, explain how it works.

Answer

We must convey two things: the suit and the rank of the hidden card.

  1. Suit: Among five cards, two share a suit by the pigeonhole principle. As a result, by sharing one of those cards we convey the suit.

  2. Rank: For the rank we must use all 4 cards to convey the information. We do this in the following way:

    • Choose which of that pair (same suit) to show so that the hidden one is at most 6 steps ahead in the cycle \(\mathrm{Ace}\to2\to\cdots\to\mathrm{King}\to\mathrm{Ace}.\) So for example, let the pair be $K\spadesuit$ and $8\spadesuit$. In this case we go 5 forward from 8 to K, hence we reveal the $8\spadesuit$.

    • There are exactly 6 permutations of the remaining three cards; pre‐agree to map each permutation to one of the distances 1–6 (through lowest to highest permutations for example). Then the other person uses that number to go forward on the rank of the first revealed card.

Now we know the suit and the rank.

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