Exploding Dots
3.2 Base Two? Base Three?
BASE TWO AND … BASE TWO AGAIN?
a) 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 this machine.
b) Write the numbers 1 through 30 as given by a \(2 \leftarrow 4\) machine and as given by a \(1 \leftarrow 2\) machine.
c) 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 \( 11 \leftarrow 3\) machine: d) Verify that a \( 11 \leftarrow 3\) machine is also a base two machine.
e) Write the numbers 1 through 30 as given by a \( 11 \leftarrow 3\) machine. Is there an easy way to the \( 11 \leftarrow 3\) representation of a number to its \( 1 \leftarrow 2\) representation, and vice versa?
FUN QUESTION: What is the “decimal” representation of the fraction \(\frac{1}{3}\) 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. This machine also converts two antidots in one box to an antidot/dot pair.
a) Show that the number \(20\) has representation \(1111\) in this machine. b) What number has representation \(1101\) in this machine? c) 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 \(1\).
d) 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. e) Suppose instead I place a \(67\) pound rock on the woman’s scale? 
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