One Rank, One Suit

The deck is not shuffled. It is built from two short lists that repeat. The values follow a cycle of thirteen: 8, K, 3, 10, 2, 7, 9, 5, Q, 4, A, 6, J, and then 8 again. The suits follow a cycle of four: clubs, hearts, spades, diamonds, and then clubs again. Each card takes the next value and the next suit.

Could a card turn up twice? Two equal cards would need a gap that is a multiple of 13 (to repeat the value) and a multiple of 4 (to repeat the suit). Since 13 and 4 share no common factor, the first gap that does both is 13 × 4 = 52. So the 52 positions hold 52 different cards: an ordinary full deck.

A cut only changes where we start reading the circle. The three dealt cards are still neighbours. The first clue gives the first value. The second clue gives the second suit, and stepping back one place on the suit cycle gives the first suit. Now the first card is known, and stepping both cycles forward names the other two.

A value and the next suit pin down where you are on both clocks at once, and the clocks then tell you every card that follows.

Step through it below. Watch the two cycles build the deck, then decode the worked example from the script: eight, then hearts.

Suits as an alphabet

Earlier tricks in this act asked for colours: an alphabet of two letters, red and black. Four colours in a row give 2 × 2 × 2 × 2 = 16 possible reports. A suit is a richer letter. It has four possible answers, so just two suits in a row already give 4 × 4 = 16 ordered reports: clubs then hearts is different from hearts then clubs, and clubs then clubs counts too.

This trick mixes two alphabets. The value has 13 possible answers and the next suit has 4, which gives 13 × 4 = 52 possible clue pairs, one for each place the deck could have been cut. Counting tells us that 52 clue pairs could be enough. The two cycles are the construction that makes every clue pair appear exactly once. A count on its own is not an arrangement: the Challenge below asks you to build one.

Set-up

1. Arrange the full deck, face down, from the top: 8♣ K♥ 3♠ 10♦ 2♣ 7♥ 9♠ 5♦ Q♣ 4♥ A♠ 6♦ J♣ 8♥ K♠ 3♦ 10♣ 2♥ 7♠ 9♦ 5♣ Q♥ 4♠ A♦ 6♣ J♥ 8♠ K♦ 3♣ 10♥ 2♠ 7♦ 9♣ 5♥ Q♠ 4♦ A♣ 6♥ J♠ 8♦ K♣ 3♥ 10♠ 2♦ 7♣ 9♥ 5♠ Q♦ 4♣ A♥ 6♠ J♦
2. You do not need to memorise 52 cards. Learn the two cycles. Values: 8, K, 3, 10, 2, 7, 9, 5, Q, 4, A, 6, J. A traditional sentence helps: “Eight kings threatened to save ninety-five queens for one sick jack.” Suits: clubs, hearts, spades, diamonds (“CHaSeD”).
3. Set out three numbered places, so that the three cards cannot be confused or reordered. A printed reference sheet of the two cycles is entirely fine while you are learning.

Performance

1. Hand over the deck. Ask for a straight cut: move a block from the top to the bottom, keeping the order inside each block. Several cuts, by several people, are fine. No shuffling.
2. Ask for the top three cards to be dealt one to each place: the first to place 1, the next to place 2, the next to place 3. Each person looks privately at their own card.
3. Place 1 names only the value (“ace, king, queen or jack, otherwise the number”). Repeat it back so you agree. Place 2 names only the suit. Place 3 says nothing.
4. Decode. Step back one suit from the second suit to get the first suit, and combine it with the named value: that is card 1. Step value and suit forward once for card 2, and once more for card 3.
5. Name the cards in order, and ask each person to turn their card over only after you have named it.

Handling warnings

• Deal to three separate places. If the three cards are dealt into one face-down pile and read from its top, their order is reversed and the decoding fails.
• Ask for rank names (“jack”), not numbers, to avoid confusion over whether a jack is eleven.
• Wraparound is not a special case: before clubs comes diamonds, and after jack comes eight.
• If the deck is shuffled, reset it. There is no decoder for a deck whose neighbour relations have been destroyed.