Overview | The Math | The Lesson

The Secret Codes

The secret codes of the lesson are closely related to the binary expansion of numbers.
Any number 0, 1, 2, 3, 4, 5 and so on, can be written as a sum of powers of 2: 2⁰ = 1; 2¹ = 2; 2² = 4; 2³ = 8; and 2⁴ = 16, 2⁵ = 32 and so on. For example: 5 = 4 + 1; 11 = 8 + 2 + 1; 8 = 8 (a power of 2 itself). This is called the binary expansion. Each number has its very own binary expansion.

Find this interesting? Read more about binary expansions.

Here we use a simplified notation — we also refer to binary expansions by the more friendly term "secret codes". Each number from 0 to 31 has a secret code consisting of some of (or all of) these numbers: 16, 8, 4, 2, 1. For example, the binary expansion of 5 is 4 + 1, but the secret code for 5 is the sequence 4, 1; the secret code for 11 is 8, 2, 1; and 8 has a secret code of just 8.

The Human Counting machine provides one way to find the secret code/binary expansion of a number.

The Human Counting Machine

The machine consists of five children labeled 1, 2, 4, 8, and 16 (as shown below).

The machine works according to these rules:

1. Each "machine component" will receive "signals" from a hand placed on his or her left shoulder. The component labeled "1" gets signals from the person operating the machine. Each of the other components of the machine will get signals from the component to his or her left.
2. When a machine component receives a signal, the reaction is:
   • machine component number 1 changes the position of the right arm every time she gets tapped on the shoulder. If the right arm was down, she lifts it and puts her hand on the shoulder of the person on her right. If her arm was already up, she lowers it.
   • machine components 2, 4, 8, and 16 follow these rules: when a hand is placed on their shoulders, they do nothing; when a hand is removed from their shoulders, they change the position of their right arm. (If the right arm was down, they lift it and put their hand on the shoulder of the person to their right. If the arm was already up, they lower it.)

The machine's output:

To find the secret code/binary expansion of a number, say 21, the operator taps 21 times on the shoulder of machine component number 1, allowing time after each tap for the machine to implement all necessary changes. The final position for 21 is:

Read off the secret code of 21 (16, 4, 1 — the labels of the raised arms) and its binary expansion (16 + 4 + 1 ).

Have a look at these other outputs of the Human Counting Machine.

How to recognize a number if you know its secret code?

Just add the numbers forming the secret code to retrieve the original number.
For example, the number whose secret code is 8, 2, 1 is revealed to be 11, since

Comments

1. The information recorded in the secret code table amounts to a dictionary for numbers from 0 to 31 and their codes.


2. You may adapt your machine to include numbers up to 63 by using an extra person labeled "32" next to the "16" component — but such a big machine wouldn't work well in practice.


3. For an amusing use of secret codes, you may look at the "Guess the Number" Magic Trick.

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