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Station  5
Explaining the 1 ← 3 Machine 

In a 131 \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 11.)

The values of individual dots throughout this machine come from noting that three ones is 33, three threes is 99, three nines is 2727, and so on.

Question 1

What is the value of a dot in the box to the left (off the screen) next after the 8181s box?
Such a dot is worth three dots in the 8181s place. Keep unexploding to move all these dots into the 11ss place. Does the count of dots you then see match your answer to the question?


Question 2

At one point we said that the 131 \leftarrow 3 machine code for the number 1515 is 120120.

Do fifteen dots in the machine indeed give one 99 and two 33s?


Question 3

What number has 131\leftarrow 3 code 2100221002?


Question 4

Stabilize this 131\leftarrow 3 machine that contains 200200 dots!


The 131\leftarrow 3 machine codes for numbers are called ternary or base three representations of numbers. Only the three symbols 00, 11, and 22 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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