Overview | The Trick | The Lesson

The Magic Trick

TEKS content: 2.12A, 3.15A, 4.14A, 5.14A.

Here is how you do the trick:

1. Ask someone in the audience to think of a number between 0 and 31 (both included). Tell the crowd you can guess the number if the person answers five yes/no questions.
    
2. Show each of the cards from your magic kit to the audience, one at a time. (The audience should see the side of the card with many numbers on it. You'll probably want to hide the fact that there is also a number on the back.)
    
3. For each card, ask, "Is the number here?" If the person who chose the number answers "yes", put the card to your right. If the student answers "no", put the card to your left.
    
4. After showing all five cards, look at all the cards that you have placed on your right, and add up the numbers on the backs of the cards. The total is the number you are trying to guess.

Have a look at this detailed example.

Why Does It Work?

TEKS Content: 2.3A, 2.13A, 2.13B, 3.3A, 3.16A, 3.16B, 4.3A, 4.15A, 4.15B, 5.3A, 5.15A, 5.15B.

The numbers on the backs of your five "magic kit" cards are 1, 2, 4, 8, and 16. These numbers are the first five powers of 2: 2⁰ = 1; 2¹ = 2; 2² = 4; 2³ = 8; and 2⁴ = 16. Each number from 0 to 31 can be written as a sum of these powers of 2. For example: 7 = 4 + 2 + 1; 21 = 16 + 4 + 1; and 8 = 8, which is itself a power of 2. This sum of powers of 2 is called a number's binary expansion. If you look at the front of the card labeled "1", you will find all the numbers from 0 to 31 that have a "1" in their binary expansion. Similarly, all the numbers that have a "2" in their binary expansion appear on the "2" card, all the numbers that have a 4 in their binary expansion appear on the "4" card, and so on. When you look at the backs of all your "yes" cards, you are actually looking at all the powers of 2 needed to decompose the secret number. Just add these powers of 2, and the mystery number is revealed!

We think children will respond more enthusiastically to the term "secret code", rather that "binary expansion". That's why we switch to secret codes. So for example, the secret code for 7 is the sequence 4, 2, 1; the secret code for 21 is 16, 4, 1; and 8 has a secret code of just 8.

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