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Station  4
Explaining The 1 ← 2 Machine 

Let’s start with the 121\leftarrow 2 machine and make sense of that curious device. Then understanding all the remaining machines will naturally follow.

Question 1

Make 22 dots explode, from the first to the second box.
Animation 2 v2 h140 opt

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Question 2

Here are four dots into the rightmost box of the machine, each worth 11.
Make them explode to get two dots in the second box (each worth 22), and then explode them!

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Question 3

If you like, you can drag 44 dots in the 11s place directly into the 44s place. Try it!

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Question 4

Here is one dot in a place we’ve labeled the 88s place. Unexplode the dot to show that it is worth two 44s. Unexplode some more to show that it is also worth four 22s. Unexplode even more to show that it is worth eight 11s.

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Question 5

Dots in the next box to the left are each worth two 88s. That is, they are worth 1616.
What are the values of dots in boxes even further to the left?

Question 6

We saw earlier that the code for thirteen in a 121\leftarrow 2 machine is 11011101.

Make those dots explode to understand this code for thirteen.

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Question 7

Make 2222 dots explode and watch the code 1011010110 appear!

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Question 8

Here are two questions you might choose to ponder.

They each require thinking of codes that require more than five boxes in a 121 \leftarrow 2 machine!

Can you use pencil and paper?

What number has 121 \leftarrow 2 code 100101100101?

Question 9

What is the 121 \leftarrow 2 code for the number two hundred?

People call the 121\leftarrow 2 codes for numbers the binary representations of numbers (with the prefix bi-meaning “two"). They are also called base two representations. One only ever uses the two symbols 00 and 11 when writing numbers in binary.

Computers are built on electrical switches that are either on or off. So it is very natural in computer science to encode all arithmetic in a code that uses only two symbols: say 11 for “on” and 00 for “off.”

Thus base two, binary, is the right base to use in computer science.

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