In this problem, you and a partner are to come up with a scheme for communicating the value of a hidden card in a normal 52-card deck. The game is played as follows:
What scheme can you and your partner use to always win?
First, choose a pair of cards of the same suit. This must be possible since you're handed more cards than there are suits. Next, pick the card in that pair that's 1–6 ranks higher than the other, if one considers ranks to wrap around, and let $n$ be the number of ranks that card is higher than the other card. This must be possible since there are only 13 cards in each suit. For instance, if the two cards are the 3 and Q of hearts, then you choose the 3 and let $n = 4$ since the 3 is four ranks higher than the Q, counting Q-K-A-2-3.
Next, hand the dealer the chosen card, followed by the other card in the pair.
Finally, compute the $n$th lowest permutation of your remaining three cards, using whatever sort order you agree on with your partner. This must be possible since there are six possible permutations and $1 \leq n \leq 6.$ Hand the dealer those three cards in the order given by that permutation.
When your partner comes in the room, she can compute $n$ based on the permutation she sees of the last three cards. She can determine the suit of the hidden card from the suit of the first face-up card. Finally, she can add $n$ to the first face-up card's rank to determine the rank of the hidden card.