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# Station  5Explaining the 1 ← 3 Machine

In a $1 \leftarrow 3$ machine, three dots in any one box are equivalent to one dot, one place their left.
(Starting with each dot in the rightmost box again worth $1$.)

The values of individual dots throughout this machine come from noting that three ones is $3$, three threes is $9$, three nines is $27$, and so on.

### Question 1

What is the value of a dot in the box to the left (off the screen) next after the $81$s box?
Such a dot is worth three dots in the $81$s place. Keep unexploding to move all these dots into the $1$ss place. Does the count of dots you then see match your answer to the question?

### Question 2

At one point we said that the $1 \leftarrow 3$ machine code for the number $15$ is $120$.

Do fifteen dots in the machine indeed give one $9$ and two $3$s?

### Question 3

What number has $1\leftarrow 3$ code $21002$?

### Question 4

Stabilize this $1\leftarrow 3$ machine that contains $200$ dots!

The $1\leftarrow 3$ machine codes for numbers are called ternary or base three representations of numbers. Only the three symbols $0$, $1$, and $2$ are ever needed to represent numbers in this system.

There is talk of building optic computers based on polarized light: either light travels in one plane, or in a perpendicular plane, or there is no light. For these computers, base three arithmetic would be the appropriate notational system to use.

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