Fitch Cheney’s Five Card Trick

The problem: An audience member selects 5 cards. The magician selects and hides one of these cards, and arranges the rest on the table. The assistant (who was absent) enters the room, looks at the four cards and states what the hidden card is. How?

We need to decide which card to hide and then want to be able to communicate what the hidden card is to our assistant. Select any two cards of the same suit from the five. (Since there are four suits, the Pigeonhole Principle tells us that at least two of the cards must be of the same suit so this will always be possible!) The hidden card must be one of these two cards.

Note that one of the two cards will be able to be obtained from the other by adding at most 6 using mod 13 addition. See the table below.

+1 / +2 / +3 / +4 / +5 / +6 / +6 / +5 / +4 / +3 / +2 / +1
Aà2 / Aà3 / Aà4 / Aà5 / Aà6 / Aà7 / 8àA / 9àA / 10àA / JàA / QàA / KàA
2à3 / 2à4 / 2à5 / 2à6 / 2à7 / 2à8 / 9à2 / 10à2 / Jà2 / Qà2 / Kà2 / Aà2
3à4 / 3à5 / 3à6 / 3à7 / 3à8 / 3à9 / 10à3 / Jà3 / Qà3 / Kà3 / Aà3 / 2à3
4à5 / 4à6 / 4à7 / 4à8 / 4à9 / 4à10 / Jà4 / Qà4 / Kà4 / Aà4 / 2à4 / 3à4
5à6 / 5à7 / 5à8 / 5à9 / 5à10 / 5àJ / Qà5 / Kà5 / Aà5 / 2à5 / 3à5 / 4à5
6à7 / 6à8 / 6à9 / 6à10 / 6àJ / 6àQ / Kà6 / Aà6 / 2à6 / 3à6 / 4à6 / 5à6
7à8 / 7à9 / 7à10 / 7àJ / 7àQ / 7àK / Aà7 / 2à7 / 3à7 / 4à7 / 5à7 / 6à7
8à9 / 8à10 / 8àJ / 8àQ / 8àK / 8àA / 2à8 / 3à8 / 4à8 / 5à8 / 6à8 / 7à8
9à10 / 9àJ / 9àQ / 9àK / 9àA / 9à2 / 3à9 / 4à9 / 5à9 / 6à9 / 7à9 / 8à9
10àJ / 10àQ / 10àK / 10àA / 10à2 / 10à3 / 4à10 / 5à10 / 6à10 / 7à10 / 8à10 / 9à10
JàQ / JàK / JàA / Jà2 / Jà3 / Jà4 / 5àJ / 6àJ / 7àJ / 8àJ / 9àJ / 10àJ
QàK / QàA / Qà2 / Qà3 / Qà4 / Qà5 / 6àQ / 7àQ / 8àQ / 9àQ / 10àQ / JàQ
KàA / Kà2 / Kà3 / Kà4 / Kà5 / Kà6 / 7àK / 8àK / 9àK / 10àK / JàK / QàK

The card on the right can be obtained from the card on the left by addition of the number at the top of the appropriate column. Therefore the card on the right is hidden and we use the card on the left as the “key card”.

The assistant will be able to tell the suit of the hidden card because it is the suit of the key card.

To lay out the 4 remaining cards, we need to communicate both the suit and the value of the hidden card.
To communicate the suit, place the key card in the mod 4 location determined by the sum of the remaining 4 cards, where we assume:
A = 1 2 = 2 3 = 3 4 = 4 5 = 5 … 10 = 10 J = 11 Q = 12 K = 13
Or mod 4 this implies
A = 1 2 = 2 3 = 3 4 = 0 5 = 1 … 10 = 2 J = 3 Q = 0 K = 1
NOTE: From left to right, the fourth card is in the 4th or 0th position!
There are six permutations of the letters L, M and H and each represents a number from 1 to 6 as follows:
LMH = 1 / LHM = 2 / MLH = 3 / MHL = 4 / HLM = 5 / HML = 6
Recall, the “key card” communicates the suit of the hidden card. It will also be used as a starting position for our count to determine the value of the hidden card.
The other three cards will determine the amount to add to the key card by using an assignment scheme so that we know which card corresponds to L, which to M, and which to H.
We use the convention of ordering the deck (Low to High) as follows:
A♣ / 2♣ / 3♣ / … / Q♣ / K♣ / A♦ / 2♦ / 3♦ / … / Q♦ / K♦ / A♥ / 2♥ / 3♥ / … / Q♥ / K♥ / A♠ / 2♠ / 3♠ / … / Q♠ / K♠
Clubs / Diamonds / Hearts / Spades
Note that the suits are in alphabetical order. This may help you to remember the convention used here!

Example 1:

2♥, 7♥, 2♦, 9♠, 7♣ are chosen. Then we must hide the 7♥. Our key card is 2♥. Our sum of the remaining 4 cards is 20 or 0 mod 4. Thus the key card must go in the 4th or last position. We need to communicate 5 more than 2♥, so we need the other 3 cards to be in the HLM order, which in this case is 9♠,7♣, 2♦ and we will therefore lay out: 9♠,7♣, 2♦, 2♥ for our accomplice to view.

Example 2:

A♥, 8♥, J♥, 6♣, 9♠ are chosen. Then we can either hide the A♥ or the J♥. We want to hide the A♥. Our key card is J♥. Our sum of the remaining 4 cards is 34 or 2 mod 4. Thus the key card must go in the 2nd position. We need to communicate 3 more than J♥, so we need the other 3 cards to be in the MLH order, which in this case is 8♥, 6♣, 9♠ and we will therefore lay out: 8♥, J♥, 6♣, 9♠ for our accomplice to view.

Reference:

http://www.spelman.edu/~colm/fitch.pdf