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## The Fitch Cheney Card Trick

Published on Sunday, August 10, 2008 in , , , ,

Imagine the following scenario, you're sitting around with several friends, and, to pass the time, one offers to show a quick card trick. The magician of the group borrows a deck of cards and asks that it be shuffled. While the shuffling is going on, the magician mentions that he is going to leave the room, and while he's gone, the person doing the shuffling is to remove any 5 cards, and hand them to another person. That person who will receive the cards is told to put the five cards in a row, with the first four in the row face-up, and the fifth one face-down. The magician then leaves the room, and the other parties do as they're told, and then call the magician back into the room. He takes one look at the arrangement of five cards on the table, and correctly names the face-down card!

This is known in magic as the Fitch Cheney Card Trick, first published by William Fitch Cheney in 1950. When done properly, this magic effect is baffling. As it happens, it's also an interesting lesson in mathematics, memory, and creative thinking. As you'll see, this trick has several layers to it, which is what makes it such an excellent lesson.

Upon first seeing the routine, many would naturally think it was marked cards. However, the deck can (and should be borrowed), so you won't always have the luxury of a marked deck. As a matter of fact, the magician can't even count on a full deck of cards, so how is this possible.

The first piece of the puzzle is that one of the people in the group is secretly helping the magician. The person who receives the cards arranges 4 of the cards in a predetermined manner, so as to communicate the identity of the 5th face-down card. However, knowing that it's up to a secret helper to arrange the cards isn't enough to understand the trick. Exactly how are the cards arranged?

What really stumps most people who get this far is that 4 cards can only be arranged in 24 different permutations (4 * 3 * 2 * 1), which isn't enough to communicate 52 different possible cards. Remember, though, that the secret helper receives 5 cards, and chooses which one will be hidden. Since 5 cards can be arranged in 120 different ways (5 * 4 * 3 * 2 * 1).

Even knowing this, you need to find a method of arranging the cards that will work regardless of what cards are given. Let's start with the suits. If you're given 5 cards, you know that you'll get at least 2 cards that have the same suit (Even if the first 4 are all different suits, the suit of the 5th card MUST match one of the first 4). This being the case, the secret helper can communicate the suit by the magician by setting one of the matched-suit cards as the face down card, and the other card of the same suit right to its left, face-up.

For example, let's say the cards are the 8S (8 of spades), 6C (6 of clubs), 10H (10 of hearts), 3D (3 of diamonds), and the 9D (9 of diamonds). In this case, either the 9D or 3D would be the face-down card, with the other diamond card right next to it.

At this point, we've determined that one card will be face-down, and that the card immediately to its left will be of the same suit. The other three cards, then need to be arranged somehow to communicate the value (the value of the cards are 1 for the ace, 11 for the Jack, 12 for the Queen, 13 for the King, and the other cards as their respective numbers). This is complicated by the fact that 3 cards have only 6 possible arrangements (3 * 2 * 1).

As long as the two cards of matching suits have values that are no more than 6 apart, such as our example cards above (3D and 9D), this is great! The other cards in our example, 8S, 6C, and 10H, could be placed in various arrangements of high (H), low (L), and medium (M) values, as long as the magician and secret helper agree on which arrangement communicates which number. A common code for this trick, based on the ternary system, is: LMH = 1 , LHM = 2, MLH = 3, MHL = 4, HLM = 5, HML = 6.

Going back to our example cards, we'd arrange the cards as 10H, 8S, 6C, 3D, and a face-down card (the 9D). When the magician comes back into the room, he sees that the first 3 cards are arranged in HML order, which means 6. With this 6 added to the 4th card, the 3D, gives the 9D, so the magician says the face-down card is the 9D!

This works well in our example, but what happens if we replace the 9D with the 10D? There's no way to communicate a 7 with the code. This is where a bit of lateral thinking comes in. What if we think of the values from 1 to 13 not as numbers in a straight line, but rather in a circle (sort of like a clock marked 1-13)? Going counterclockwise from 10 to 3 would take 7 steps, but going clockwise from 10 to 3 only takes 6 steps (this is modular arithmetic)!

So, using the “clock approach” above, we would arrange the cards as 10H, 8S, 6C, 10D, and the 3D face-down. As before, the HML arrangement codes the number 6, but this time the 6 is added to the 10D. The magician would think of this as the “16D”, and then subtract 13, giving the 3D, and announces this as the answer. As you can see, this 1 through 6 code can be used to secretly communicate the difference between any two values.

There is one remaining complication. While it may be unlikely, let's take the worst case scenario of 3D, 3C, 3H, 3S, and the 9H into consideration. As before, the 9H will be used to communicate the heart suit, and as a starting point to get to the 3. However, this leaves us with the 3D, 3C, and 3S. How do you communicate low, medium, and high when all the cards are 3s? Even just two cards of the same value would present a similar problem.

Fortunately, a classic magician's tool, known as the CHaSeD order, comes in handy here. Instead of just using the card's values to determine whether it's high, low, or medium, the suits are also used. CHaSeD refers to a specific order of suits: Clubs, followed by Hearts, followed by Spades, then followed by Diamonds (This suit order has no particular value over other order, except that it's already well-known among magicians).

Using this order, we'll declare that clubs have the lowest “value” of all the suits, with hearts having a higher “value” than clubs, spades having a higher “value” than hearts and clubs, and diamonds having a higher “value” than any other suit. In this system, a 2S would be considered as having a higher value than the 2C, because spades are considered as having a higher value than clubs. Keep in mind, this system of suit values is only employed when 2 ore more cards are of the same value. Even in this system, you'll never have to deal with questions like whether the KH is considered to be higher or lower than the 2S.

Gong back to out latter example, we'd arrange the cards as 3D, 3S, 3C, 3H, and the 9H face-down. This would definitely throw anybody off the trail who is trying to reverse-engineer the routine. The magician, however, sees the 3D, followed by the 3S and the 3C, and using his knowledge of the CHaSeD order, is able to work out that this is an HML order, coding 6. The 6 is added to the 3H, giving the magician the coded answer of 9H.

Obviously, this will take regular practice between the 2 people, in order to make sure that all situations can be handled smoothly, the cards arranged and coded correctly, and the performance refined so that the secret communication between the two people remains a secret.

Imagine the work it must've taken to think of this in the first place, especially considering the codes, mathematics, and memory involved. The efficient and ingenious method, combined with the simple appearance of the effect to the audience, quickly made the Fitch Cheney Card Trick a classic.

Many magicians have even taken it as a challenge to figure out how to minimize the number of cards used in this routine, while still keeping the code employed easy enough to put into effect with minimal hesitation, both on the part of the secret helper doing the coding, and the magician, when doing the decoding. Colm Mulcahy, in his Card Colm column, published a version called Fitch Four Glory that involves the use of only 4 cards!

### 3 Response to The Fitch Cheney Card Trick

6:19 AM

can we communicate negative distances in this trick. that is 3D faced down and 9D faced up as in your example.

12:06 AM

Here is another method posted on Quora by Ori Eyal:

Pre-arrange to number the cards from 1 to 52 according to suit and number. Then
four cards can be arranged to represent a number from 1 to 24. Now determine the
values for the five cards. If the highest card is less than 24 away from the
second-highest, use it as the secret card, otherwise use the lowest card as the
secret card. Even if the highest card is 25 higher than the second-highest, the
lowest card must then be at least 3 lower than the second-highest, i.e. at least
28 lower than the highest; which places it at a distance of 23 from the highest
card when counting up and wrapping around.

Curiosity
7:15 AM

In the extreme case, how would one encode a suit using a flush of different suit?