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BASE TWO AND BASE TWO

  1. Verify that a 242 \leftarrow 4 machine is a base-two machine. (That is, explain why x=2x = 2 is the appropriate value for xx in the picture below.)
    I9S9C - Image115

  2. Write the numbers 11 through 3030 as given by a 242 \leftarrow 4 machine and as given by a 121 \leftarrow 2 machine.

  3. Does there seem to be an easy way to convert from one representation of a number to the other?

(Explore representations in 363 \leftarrow 6 and 5105 \leftarrow 10 machines too?)

Now consider a 1131|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!)
I9S9C - Image121

  1. Verify that a 1131|1 \leftarrow 3 machine is also a base two machine.
    I9S9C - Image139

  2. Write the numbers 11 through 3030 as given by a 1131|1 \leftarrow 3 machine. Is there an easy way to convert the 1131|1 \leftarrow 3 representation of a number to its 121 \leftarrow 2 representation, and vice versa?

FUN QUESTION: What is the “decimal” representation of the fraction 13\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 11021|-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

  1. Show that the number twenty has representation 11111|-1|1|-1 in this machine.
  2. What number has representation 11011|1|0|-1 in this machine?
  3. This machine is a base machine:
    I9S9C - Image128

Explain why xx equals 33.
Thus the 11021|-1 \leftarrow 0|2 machine shows that every number can be written as a combination of powers of three using the coefficients 11, 00 and 1-1.

  1. A woman has a simple balance scale and five stones of weights 1,3,9,271, 3, 9, 27 and 8181 pounds.
    I9S9C - Image138

I place a rock of weight 2020 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.

  1. Suppose instead I place a 6767 pound rock on the woman’s scale. Can she make that stone balance too?
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