Overview | The Math | The Lesson

Number Path #1

Take a look at this example.

Step by step, a number is modified through addition, subtraction and multiplication. It turns out that it always ends up just being multiplied by 10. Here is a formal explanation of that fact. It requires the use of the distributive law.

Number Path #2

Step by step, a number is modified through addition, subtraction and multiplication. The end result turns out to always be 20 times the original number + 1. This fact may also be established in a formal way. The work is very similar to that done for Number Path #1.

Blank Number Path

Some magicians build magic tricks based on human Number Paths. Let's see how this is done if the magician uses Number Path #1. Somebody in the audience is asked to think of a secret number and whisper it to the next person. This person adds 2 and whispers the result to somebody else. This person multiplies by 5, the next person subtracts 7, the next multiplies by 2, and the last person subtracts 6. The resulting number is announced aloud. The magician, upon hearing this number, quickly divides it by 10 and guesses the secret number.

Now, since dividing by 10 is so easy, this might not be the most impressive magic trick. But if you use another Number Path (say, one whose shortcut is to multiply by 5), you'll have a better magic trick. Of course, you'll need to be very fast in dividing numbers by 5 mentally.

The Blank Number Path provides a way to construct a path whose shortcut is prescribed — for example, to multiply by 5, or to multiply by 6 and subtract 2, etc. For young participants, constructing a Number Path with a pre-assigned shortcut will be an exercise in trial and error.

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