# The Five-card Trick

The Great Marvo demonstrates his latest trick. First his assistant, the lovely Glitzy Mitzy, presents a pack of cards to a member of the audience, who picks out five cards at random and gives them to Mitzy. She looks at the cards and then lays them out on the table; four are face upward but the fifth is face downward so that its identity is hidden. Now the Great Marvo demonstrates his magic - using his mind-reading abilities, he reads the identity of the fifth card from the air and announces it to an amazed audience. Mitzy turns over the card and, sure enough, Marvo is right!

### How It's Not Done

How does Marvo figure out the identity of the hidden card? It's not done by Mitzy winking, coughing or tapping her nose to give signals to Marvo. In fact, Mitzy can be sent out of the room once she has placed the cards on the table and Marvo will still identify the card. The pack of cards is a normal one, not marked in any way. There are no mirrors involved, no hidden cameras, and no supernatural powers.

### How It Is Done

Mitzy does most of the work in this trick. She chooses which one of the five cards is to be the hidden one, and arranges the other four in a way which gives coded information to Marvo. Marvo has the easier task. He just looks at the four visible cards and decodes the identity of the hidden card.

#### How Marvo Decodes the Information

Firstly, the suit of the hidden card is the same as that of the first card. If the first card is a seven of spades, then Marvo knows the hidden card is a spade.

Now Marvo looks at the other three visible cards. One card is the highest of the three, one is the lowest and one is in between. Marvo works out what order they are in (for example: Medium, Low, High) and converts this to a number using the following table:

• Low Medium High = +1
• Low High Medium = +2
• Medium Low High = +3
• Medium High Low = +4
• High Low Medium = +5
• High Medium Low = +6

Now Marvo counts up from the first card by this number until he reaches the number of the hidden card. The order of the cards is Ace up the numbers to Jack, Queen, King, then back to Ace and up the numbers again. If the first card is the seven of spades and his number is +6, he counts from seven up to King. The hidden card is the King of Spades.

#### A Boring but Important Detail

The question arises: what does Marvo do if two of the cards have the same numerical value? For example, there might be a five of clubs and a five of hearts. Marvo and Mitzy have agreed in advance an order for the suits. Because they are Bridge players, they like to use 'Spades, Hearts, Diamonds, Clubs', with spades ranking highest. In this order, spades are higher than hearts, which are higher than diamonds, which are higher than clubs. So the five of hearts is higher than the five of clubs. Any suit order will do as long as it is agreed in advance.

#### How Mitzy Lays out the Cards

Mitzy must carry out the following steps:

1. Pick two cards from the five that are in the same suit. Since there are five cards, at least two will always be in the same suit.

2. Decide which of the two is to be the hidden card. The hidden card must be six or less above the other card, remembering to wrap around from King to Ace to two. If it is more than six, then make the other card the hidden card. Since there are only 13 cards in a suit, this will always be possible.

3. Encode this number into a pattern of Low, Medium and High using the table.

4. Lay the cards out.

5. Remember to let the Great Marvo take all the credit, while grinning inanely.

### A Mathematical Trick

The trick was invented by US mathematician William Fitch Cheney Jr (1894 - 1974). It was first published in 1950, in Maths Miracles by Wallace Lee.

At first sight it is mystifying, because the hidden card can be any one of 48 - in fact, any card in the pack other than the four that are visible. The identity of the card is hidden in the arrangement of the other four cards, which can be arranged in only 24 different ways. So there seems to be not enough information to allow Marvo to identify the card.

The answer is that Mitzy herself chooses the card to be hidden from among the five given to her, and as the magician now knows its suit, there are only 12 or fewer cards to identify it from. The trick could theoretically be done with a much bigger pack, although the exact details would have to be changed slightly - the size of the pack depending on how many details we change. For example, allowing Mitzy to place the face-down card among the other cards rather than always at the start opens up the possibilities enormously.

Of course, this is more of interest to mathematicians than to stage magicians, who could perform a similar trick in many different ways, such as by forcing the audience to choose a particular set of five cards in the first place.