Station 4
Explaining The 1 ← 2 Machine

Let’s start with the $1\leftarrow 2$ machine and make sense of that curious device. Then understanding all the remaining machines will naturally follow.

Let’s go!

Question 1

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

Question 2

Here are four dots into the rightmost box of the machine, each worth $1$ .
Make them explode to get two dots in the second box (each worth $2$ ), and then explode them!

Question 3

If you like, you can drag $4$ dots in the $1$ s place directly into the $4$ s place. Try it!

Question 4

Here is one dot in a place we’ve labeled the $8$ s place. Unexplode the dot to show that it is worth two $4$ s. Unexplode some more to show that it is also worth four $2$ s. Unexplode even more to show that it is worth eight $1$ s.

Question 5

Dots in the next box to the left are each worth two $8$ s. That is, they are worth $16$ .
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 $1\leftarrow 2$ machine is $1101$ .

Make those dots explode to understand this code for thirteen.

Question 7

Make $22$ dots explode and watch the code $10110$ appear!

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 $1 \leftarrow 2$ machine!

Can you use pencil and paper?

What number has $1 \leftarrow 2$ code $100101$ ?

Question 9

What is the $1 \leftarrow 2$ code for the number two hundred?

People call the $1\leftarrow 2$ 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 $0$ and $1$ 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 $1$ for “on” and $0$ for “off.”

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