BASE TWO AND BASE TWO

Verify that a $2 \leftarrow 4$ machine is a base-two machine. (That is, explain why $x = 2$ is the appropriate value for $x$ in the picture below.)
Write the numbers $1$ through $30$ as given by a $2 \leftarrow 4$ machine and as given by a $1 \leftarrow 2$ machine.
Does there seem to be an easy way to convert from one representation of a number to the other?

(Explore representations in $3 \leftarrow 6$ and $5 \leftarrow 10$ machines too?)

Now consider a $1|1 \leftarrow 3$ machine. Here three dots in a box are replaced by two dots: one in the original box and one one place to the left. (Weird!)

Verify that a $1|1 \leftarrow 3$ machine is also a base two machine.
Write the numbers $1$ through $30$ as given by a $1|1 \leftarrow 3$ machine. Is there an easy way to convert the $1|1 \leftarrow 3$ representation of a number to its $1 \leftarrow 2$ representation, and vice versa?

FUN QUESTION: What is the “decimal” representation of the fraction $\dfrac{1}{3}$ in each of these machines?
How does long division work for these machines?

A DIFFERENT BASE THREE

Here’s a new type of base machine. It is called a $1|-1 \leftarrow 0|2$ machine and operates by converting any two dots in one box into an antidot in that box and a proper dot one place to the left. It also converts two antidots in one box to an antidot/dot pair.
I9S9C - Image125

Show that the number twenty has representation $1|-1|1|-1$ in this machine.
What number has representation $1|1|0|-1$ in this machine?
This machine is a base machine:

Explain why $x$ equals $3$ .
Thus the $1|-1 \leftarrow 0|2$ machine shows that every number can be written as a combination of powers of three using the coefficients $1$ , $0$ and $-1$ .

A woman has a simple balance scale and five stones of weights $1, 3, 9, 27$ and $81$ pounds.

I place a rock of weight $20$ pounds on one side of the scale. Explain how the women can place some, or all, of her stones on the scale so as to make it balance.