The 5 cards given to the first player must contain two cards of the same suit. Of these, one must be six or less greater than the other, wrapping around from K to A to 2. For example, suppose that there are two spades, the 5 and the J. The spades which are six or less greater than the J are the Q, K, A, 2, 3 and 4, and the 5 is not among these; but the spades which are six or less greater than the 5 are the 6, 7, 8, 9, 10 and J, and the J is among these. Thus of the two spades, the J is the one which is six or less greater than the 5.

The first player then keeps the card which is six or less greater than the other (in the example, the J of spades) and shows the other (in the example, the 5 of spades) as the first (leftmost) of the 4 exposed cards. The remaining cards are considered ordered by rank, with A highest and 2 lowest, and among cards of equal rank in the order SHDC, with spades(S) highest and clubs(C) lowest. The first player shows the remaining three cards in one of six permutations according to this ordering, corresponding to the numbers 1 to 6, whichever of these is the difference between the hidden card and the one already shown. This means, if the three cards are called L (for low), M (for middle) and H (for high) according to the ordering, then they should be

placed in one of the following orders
1 LMH 2 MLH 3 LHM 4 HLM 5 MHL 6 HML

thus in the example, he would place them in the order HML, since the J of spades is 6 greater than the 5.

When the second player returns to the room, he knows that the hidden card is of the same suit as the leftmost card, and he adds to the rank of this card the number 1 to 6 corresponding to the permutation of the remaining three cards, thus identifying the hidden card uniquely. In the example, the second player sees

5S H M L

and since HML corresponds to 6, he adds 6 to the 5 of spades to get the J of spades.

Of course, the first player may have more than one choice, since there might be three cards of the same suit, or two cards each of two different suits, but he simply chooses one pair of cards of the same suit, and the fact that there are other choices is irrelevant.

See also URLs: http://www.anamorph.com/docs/ct/cards.html http://www.frontiernet.net/jmb184/interests/puzzles/4_of_5_trick.html

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