How to do the Fitch Cheney five-card trick

A spectator picks any five cards and hands them to your partner. Your partner shows you four of them, one at a time, and you name the fifth. There are no marks or gimmicks: the order your partner shows the cards is a code. The first card tells you the suit, and the order of the other three tells you how far to count. It was first mentioned in Math Miracles by Wallace Lee in 1950, who credits it to William Fitch Cheney Jr.

What does the audience see?

Someone shuffles a normal deck and chooses any five cards. They hand them to your assistant, who looks at them and passes four back to you, one at a time. You study the four cards for a moment, then name the card your assistant is still holding.

What do you need?

• An ordinary deck of 52 cards.
• A partner who knows the same system. Agree in advance whether aces count as low; this guide counts the ace as 1 and the king as 13.

Step 1: How does your partner pick the hidden card?

With five cards and only four suits, at least two cards always share a suit. Your partner looks for that pair. One card of the pair will be hidden, and the other will be shown to you first, so it tells you the suit.

Step 2: Which card of the pair stays hidden?

Picture the 13 ranks arranged in a circle like a clock: A, 2, 3 … 10, J, Q, K, then back to A. From one card of the pair, you can always reach the other by counting clockwise 6 steps or fewer. The card you count from is the one shown first. The card you count to stays hidden.

For example, with a 10 and a 2 of the same suit, count clockwise from the 10: J, Q, K, A, 2. That is 5 steps, so the 10 is shown and the 2 is hidden. Counting from the 2 up to the 10 would take 8 steps, which is too many.

Step 3: How do the other three cards give the number?

Your partner now has to tell you a number from 1 to 6 using the three remaining cards. First, rank them as low, middle and high. Compare ranks first; if two cards share a rank, use bridge order for the suits: clubs, then diamonds, then hearts, then spades (lowest to highest). Then place them in the order that matches the number:

• Low, middle, high = 1
• Low, high, middle = 2
• Middle, low, high = 3
• Middle, high, low = 4
• High, low, middle = 5
• High, middle, low = 6

The full sequence is the counting card first, then the three code cards in the right order.

How do you work out the hidden card?

1. Read the suit. The hidden card is the same suit as the first card shown.
2. Read the number. Sort the next three cards into low, middle and high in your head and see which order they came in.
3. Count. Count that many steps clockwise from the first card’s rank, wrapping from king to ace.

Can you walk through an example?

Mathematician Michael Kleber opens his article on the trick with this one. Your partner hands you the 7♠, then the Q♥, the 8♣ and the 3♦.

• The first card is a spade, so the hidden card is a spade.
• Of the other three, the 3♦ is low, the 8♣ is middle and the Q♥ is high. They came in the order high, middle, low, which means 6.
• Six steps on from the 7 is the king. The hidden card is the K♠.

How do you make it look smooth?

• Practise the code together. Deal five cards, let your partner encode them, and decode them out loud until neither of you has to think about it.
• Let the spectator shuffle. The method works with any five cards, so a fully shuffled deck makes the result stronger.
• Take your time. A short pause before you name the card makes it feel like mind reading rather than arithmetic.

Where does the trick come from?

It was first mentioned in Math Miracles by Wallace Lee in 1950, who credits it to William Fitch Cheney Jr., known as “Fitch”. Kleber notes that he doesn’t know whether it appeared earlier. Cheney was a mathematician, and the sources we read describe him as holding the first math PhD awarded by MIT.

Kleber’s article, “The Best Card Trick”, appeared in The Mathematical Intelligencer in 2002.

What helps you take it further?

• Self Working Card Tricks by Karl Fulves. 72 card tricks that, as its description puts it, work automatically because of the mathematics in the deck itself. Almost all can be done informally with a borrowed deck.
• Mathematics, Magic & Mystery by Martin Gardner. A study of the math behind tricks with cards, dice and coins, with more than 500 tricks that need no sleight of hand.

For more card tricks that run on math, try the 21 card trick and the magic number nine trick, or browse our playing cards.

Frequently asked questions