[This is my first Instructable! I hope you enjoy!]
This is a cute trick that I learned many years ago. If you have nephews or nieces, this trick can get you that coveted "favorite uncle" status, and MAKE YOUR SIBLINGS JEALOUS.
It involves a little bit of math for the audience member that's choosing the "card", so make sure that they're old enough to multiply.
The true artistry of this trick is hamming up the act with the "invisible deck of cards" to make it look like you're an expert shuffler, and all the ways you can make the audience laugh along the way. If you have any more suggestions on how to make it look even more over the top, please let me know in the comments; I mention my favorites here, but I'm always happy to hear of others.
Step 1: The Set Up
Announce to the onlookers:
I've got a card trick, and it's so good, I don't even need any real cards to do it! Any volunteers?
Make a big show out of pulling an imaginary deck of cards from your pocket, and start moving your hands as if you were shuffling, throwing them all in the air and catching them behind your back, and then ask your volunteer to cut the deck. Ask them if they would like to shuffle the cards as well, then tell them to pick up the cards they dropped on the floor.
Now fan out your cards in front of you, and ask them to pick one out. After they do so, throw the entire deck of cards up into the air, look up and wait a couple of seconds, then open your shirt or pants pocket to catch them all.
Step 2: The Trick
Instructions to your volunteer:
Imagine that what you have in your hand is a real card from a regular playing card deck. No jokers, and no making up a new card like the Crab of Ineffable Wisdom.
Your card has a value and a suit, right? Now take the value of the card, and multiply it by 2.
[If it looks like they're in trouble, gently remind them that a Jack is 11, Q=12, K=13, and an Ace can be either 1 or 14, but it'll be easier if they stick to 1]
Got it? Now, add 1.
Still with me? OK, this is the hard part. Multiply your total by 5.
[This is what tends to trip up most people, but give them confidence that they can do it.]
Now, I'm going to ask you to add something to your total based on the suit.
If your card is a Club, add 6;
if it's a Heart, add 7;
for Spades, add 8;
and for Diamonds, add 9.
[ The easy way to remember this is to use the mnemonic word CHaSeD for clubs, hearts, spades, diamonds]
All done? Then tell me what your total is.
[ For this example, let's assume they tell you their total is 33. After looking them, and making a few funny faces about reading their mind, let them know]
Your card must be the Two of Spades!
[After they're amazed, and they start to go back to their seat. . . ]
Hey, give me my card back!
Step 3: The Solution
The number that your volunteer gives you will either be a two digit number or a three digit number. The important part is to split it in two pieces, with the ones place on the right.
Volunteer's total: 33
3 // 3 (ones place on the right of the // separator, and any other digits on the left)
Take the number on the left side and subtract one; that's the Value of their card.
Use the same order for the last step in the trick to process the right side:
If it's a 1, their card was a Club
If it's a 2, their card was a Heart
A 3 points to Spades,
and a 4 is a Diamond.
Put it all together:
3 // 3
Value of the card: 3 - 1 = 2
Suit: 3 -> Spades
The Two of Spades!
Another example, let's say that the total they gave you was 134.
Split it in 2, with the ones place on the right, and all other digits to the left.
13 // 4
Value of the card: 13 - 1 = 12 (Queen)
Suit: 4 -> Diamonds
The Queen of Diamonds!
If the last digit of their total isn't a one, two three or four, then they must have done something wrong. Accuse them (jokingly) of making up a card.
Were you thinking of the Zebra of Silliness? You know that's not a real card! Let's start over. . .
Step 4: The Math
What this trick is doing is setting up a very simple pairing function (see NOTE below), which combines two values to make one total, and that total is de-composable to get back the original two values.
Let's do a little bit of algebra to demonstrate. Assume you have the value of the card as a variable V, and the suit will be a variable S, which has to be a number from 6 to 9 based on the trick as you explained it.
We told the volunteer to multiply the value of the card by 2, add 1, then multiply by 5.
Partial total: (2*V + 1) * 5
== 5(2V + 1)
== 10V + 5
So, all you're doing with the value is multiplying it by 10 and then adding 5.
An Ace will always give you a partial total of 15, a Two will be 25, a Three 35, etc.
Then adding S (remember, a value between 6 and 9) will make the tens place go up by one, and the ones place will become a value equal to S - 5.
That's really all there is to it. But since everyone knows their tens tables, hiding the process using multiple steps (*2, +1, *5) makes it all more mysterious and cryptic, and makes your audience think that the process is much more complicated than it really is.
- - - -
NOTE: The algebra behind this trick is similar to a pairing function, but a true pairing function can combine numbers much larger than the small range of values and suits in a card deck. In fact, any two natural numbers (whole numbers starting from 0, 1, etc.) can be combined to make a larger number, and that larger number has a unique way of being decomposed into the original numbers again.