Easy Card Tricks – a Magician’s Delight

This easy card trick is based on the mathematical principle of information theory. It is very easy to perform but hard to figure out – so it leaves audiences of all ages in awe!

Needed for the easy card trick:
1. A magician
2. An assistant to the magician
3. A deck of 52 cards (no jokers)
4. An audience

Easy card tricks: Step-1
In the absence of the magician, the assistant shuffles the deck of 52 cards (without jokers) and asks the audience to randomly select 5 cards from the deck. Up to 5 people from the audience may be asked to pick the cards. The selected cards are shown to all in the audience and handed over to the assistant.

Easy card tricks: Step-2
The assistant returns a specific card to the audience and rearranges the remaining 4 cards in a certain order. The magician is called and given the 4 cards. Lo and behold! The magician calls the returned card correctly !

Easy Card Tricks: Underlying Idea
The trick lies in how the assistant selects the card to return to the audience and how he rearranges the other cards based on a predetermined plan to convey information about the returned card to the magician. Below we describe the logic and provide some examples.

Easy Card Tricks: Notation:
Let S,C,D,H denote the 4 suits – spades,clubs, diamonds and hearts respectively. Below, cards are denoted by their number and suit. For example Jack Spades is denoted by JS.

Easy Card Tricks: Three Definitions
To perform the trick, the assistant and magician have a common understanding of the following.

Card-Rank: A relative rank is assigned to each card based on its suit and number. When comparing 2 cards with different numbers, a card with a larger number has a higher rank. If the 2 cards have the same number but belong to different suits, we follow the convention that S has the highest rank followed by C, D and H in descending order.

Example:
Between 8H and 3S, 8H has a higher rank
Between 4C and 4S, 4S has a higher rank

Card-Distance: Let the cards from A,2…J,Q,K represent the 13 numbers 1,2,…11,12,13. Now arrange them in a circle (like in an analog clock) clock-wise so that K (13) is adjacent to A (1) as shown below:

………….Q….K
…….J……………A
…..10………………2
….9…………………..3
……8………………..4
…….7……………..5
……………..6

Equivalently in numbers:

…………12….13
…….11…………..1
…..10……………….2
….9……………………3
……8…………………4
…….7……………..5
……………..6

The card-distance between any two cards X and Y is the number of locations counted along the circle to move from card X to card Y in a clock-wise manner.

Example:
Card-distance between 2D and 8D is 6.
Card-distance between 8D and 2D is 7.
Card-distance between KH and AH is 1.
Card-distance between AH and KH is 12.

Mapping-Table:
When comparing the ranks of three cards, let G, M and L denote the greatest, medium and lowest ranks. Then, the following table maps the rank of 3 cards arranged in all possible combinations of their ranks. This table is memorized by the magician and his assistant.

Index……….Rank Arrangement
1…………………….. G,M,L
2…………………….. G,L,M
3…………………….. M,G,L
4…………………….. M,L,G
5…………………….. L,G,M
6…………………….. L,M,G

Easy Card Tricks: Which card to return to the audience?
Of the five cards, return a card that satisfies both of the following:
a. It is from a suit of which 2 or more cards are present amongst the 5 chosen cards. Call the suit of the returned card “return-suit”.
b. The card-distance between another card from the return-suit and the returned card is at most 6. Call this other card the “hint-card”.

Easy Card Tricks: How to arrange the remaining 4 cards?
When rearranging the remaining 4 cards, the assistant places the hint-card right on top. So immediately, the magician knows that the suit of the returned card is same as that of the hint-card (return-suit). The remaining 3 cards are arranged so that their card-ranks convey the distance of the returned card from the hint-card; this is the the index from the mapping-table. The magician simply moves index positions from the hint-card clock-wise and calls out the number of returned card.

Example 1:
Step-1: Assume audience picks cards: 3D, 7D, 5S, 6S, 10H
Step-2: Card to return to audience: 7D, Cards to hand to magician (top to bottom): 3D, 6S, 5S, 10H. Magician calls 7D.

How the magician infers the returned card:
Suit of the returned card (return-suit) = D
Index from mapping-table for the three remaining cards 6S, 5S, 10H with ranks M,L,G = 4; number of the returned card = 3 + index = 4

Example 2:
Step-1: Assume audience picks cards: 10D, 10S, AC, 5H, KH
Step-2: Card to return to audience: 5H, Cards to hand to magician (top to bottom): KH, AC, 10S, 10D. Magician calls 5H.

How the magician infers the returned card:
Suit of the returned card (return-suit) = H.
Index from mapping-table for the three remaining cards AC, 10S, 10D (with ranks L,G,M) = 6; number of the returned card = K + index = 5.