## Exploding Dots

### 3.1 On Base One-and-a-Half?

Lesson materials located below the video overview.

Recall the $$2 \leftarrow 3$$ machine from section 1. It takes three dots in one box and replaces them with two dots one place to the left.

What is this machine? Is it a base machine?

Well the title of this section gives it away, but let’s reason our way through the mathematics of this machine.

Dots in the rightmost box, as always, are each worth “$$1$$.” And three of these dots are equivalent to two dots one place to the left.

If the dots in this second box are each worth $$x$$ then we have $$3 \cdot 1 = 2 \cdot x$$  giving: $$x=\dfrac{3}{2}$$.

We also see that $$3 \cdot \frac{3}{2} = 2y$$. (Here $$y$$ is the value of the next box to the left.)

This gives: $$y = \dfrac{3}{2} \cdot \dfrac{3}{2} = \left(\dfrac{3}{2}\right) ^ {2}$$.

Also $$3 \cdot \left(\frac{3}{2}\right)^{2} = 2z$$  (with $$z$$ the value of the next box to the left).

This gives $$z = \frac{3}{2} \cdot \left(\frac{3}{2}\right)^{2} = \left(\frac{3}{2}\right)^{3}$$.

And so on!

This machine produces the base $$\frac{3}{2}$$ representations of numbers! Welcome to a system that one could call base one-and-a-half! This is a version that uses the digits $$0$$, $$1$$, and $$2$$.

Comment: Members of the mathematics community might prefer not to call this base one-and-a-half in a technical sense as we are using the digit “2” in our representations. This is larger than the base number. To see the language and work currently being done in this field, look up beta expansions and non-integer representations on the internet. In the meantime, understand that when I refer to “base one-and-a-half” in these notes I really mean “the representation of integers as sums of powers of one-and-a-half using the coefficients 0, 1, and 2.” That is, I am referring to the mathematical representations that arise from $$2 \leftarrow 3$$ machines.

The $$2 \leftarrow 3$$ machine representations of the first twenty-four numbers are:

Comment: I personally find this alarming! Each representation listed here represents a combination of some very strange fractions, the fractions $$1$$, $$\frac{3}{2}$$, $$\frac{9}{4}$$, $$\frac{27}{8}$$, $$\frac{81}{16}$$, and so on. Yet these combinations are meant to fit together perfectly each time to produce whole number answers! Consider the number ten, for example. It has representation: $$2101$$.

Is it true that $$2 \times \frac{27}{8} + 1 \times \frac{9}{4} + 0 \times \frac{3}{2} + 1 \times 1$$ is the perfect whole number ten?

Whoa!

There are plenty of questions to be asked about numbers in this $$2 \leftarrow 3$$ machines version of base one-and-a-half, and many link with unsolved research issues of today!

 QUESTION 1:  PATTERNS? Are there any interesting patterns to these representations? Why must all the representations (after the first) begin with the digit $$2$$? Do all the representations six and beyond begin with $$21$$? If you go along the list far enough do the first three digits of the numbers become “stable”? The first four?   What can you say about final digits? Last two final digits?   Comment: Dr. Jim Propp of UMass Lowell, who opened my eyes to the $$2 \leftarrow 3$$ machine, suggests these more robust questions. What sequences can appear at the beginning of infinitely many $$2 \leftarrow 3$$ codes? What sequences can appear at the end of infinitely many $$2 \leftarrow 3$$ codes? What sequences can appear somewhere in the middle of infinitely many $$2 \leftarrow 3$$ codes?

 QUESTION 2:  UNIQUENESS Prove that these $$2 \leftarrow 3$$ machine codes for numbers using the digits $$0$$, $$1$$, and $$2$$ are unique. [Also… Prove that every whole number can be written – uniquely – as sums of powers of $$\frac{7}{5}$$ using the coefficients $$0,1,2,3,4$$. And that every whole number can be written – uniquely – as sums of powers of $$\frac{13}{8}$$  using the coefficients $$0,1,2,3,4,5,6,7$$. And that every whole number can be written – uniquely – as sums of powers of $$\frac{339}{56}$$ using the coefficients $$0,1,2,…,54,55$$. And so on!]

 QUESTION 3:  EVEN NUMBERS? (I DON’T KNOW THE ANSWER!) Here are the first forty numbers written their $$2 \leftarrow 3$$ machine code (and zero at the start).   Since the final digits cycle “$$0,1,2$$” we  see the following divisibility rule for three:   A NUMBER WRITTEN IN $$2 \leftarrow 3$$ CODE IS DIVISIBLE BY THREE PRECISELY WHEN ITS FINAL DIGIT IS ZERO.   What’s a divisibility rule for two? How can you tell if a number is even when it is written in its $$2 \leftarrow 3$$ code? What common feature does every second code have?

 QUESTION 4:  IS IT AN INTEGER? (I DON’T KNOW THE ANSWER!) Here are the first forty numbers written in their $$2 \leftarrow 3$$ machine codes again.   Not every collection of $$0$$s, $$1$$s, and $$2$$s will represent a whole number in a $$2 \leftarrow 3$$ machine. For example, is: $$210221202012002012201102202010221021200202212$$ a whole number? How can one tell if a given representation corresponds to an integer?   Of course, one can just compute the sum of fractions a given sequence of digits represents and see if that sum is an integer. But is there a more efficient, “quick,” means of identifying when such a sum will be a whole number? (Of course, what we mean by quick and efficient can be debated!)

 QUESTION 5:  NUMBER OF DIGITS  (I DON’T KNOW THE ANSWER!)     Notice: $$0$$ dots gives the first one-digit answer (Some might prefer to say $$1$$ here.) $$3$$ dots gives the first two-digit answer $$6$$ dots give the first three-digit answer $$9$$ dots give the first four-digit answer and so on.   This gives the sequence:  3, 6, 9, 15, 24, …. (Let’s skip the questionable start.) Any patterns?   COMMENT: Are you thinking Fibonacci? Sadly the next few numbers are $$36, 54, 81, 123, 186, 279, 420, 630, …$$.   A recursive formula. Let $$a_{N}$$ represent the $$N$$th number in this sequence, and regard $$1$$ -not zero- as the first one digit answer. It is known that:   $$a_{N+1} = \dfrac{3a_{N}}{2}$$ if $$a_{N}$$ is even. $$a_{N+1} = \dfrac{3\left(a_{N}+1\right)}{2}$$ if $$a_{N}$$ is odd.   (If $$m$$ dots are needed in the right most box to get an $$N$$-digit answer, how many dots are needed to get $$m$$ dots in the second box?)   An explicit formula? Is there an explicit formula for $$a_{N}$$? Is it possible to compute $$a_{100}$$ without having to compute $$a_{99}$$ and $$a_{98}$$ and so on before it?   Comment: This question was first posed by Dr. James Propp.

RELATED ASIDE?? In 1937 L. Collatz was playing with numbers and proposed the following amusement:

Choose any positive integer.

If it is even, divide it by two.

If it is odd, triple it and add one.

Either way, you now have a new integer.

Repeat.

For example, starting with $$7$$ we obtain the sequence:

$$7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52 \rightarrow 26$$

$$\rightarrow 13 \rightarrow 40 \rightarrow 20 \rightarrow 10 \rightarrow 5 \rightarrow 16$$

$$\rightarrow 8 \rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow 4 \rightarrow 2 \rightarrow 1$$

$$\rightarrow 4 \rightarrow 2 \rightarrow 1 \rightarrow \cdots$$

Try this game for yourself with different starting integers. Do they too fall into a 4-2-1 cycle? (Try starting with $$27$$.)

Does every integer eventually lead to a $$4 \rightarrow 2 \rightarrow 1$$ cycle?

No one currently knows … but the conjecture has been checked and found to hold for all numbers from $$1$$ through to $$3 \cdot 2^{53}$$ (which is about 270000000000000000).

Comment: This problem is sometimes called the HAILSTONE PROBLEM because the chain of numbers one obtains seems to bounce up and down for a good while before eventually falling down to 1. For example, the number 27 takes 112 steps before entering a $$4 \rightarrow 2 \rightarrow 1$$ cycle, reaching a high of 9232 before getting there!

a)    Play with Collatz conjecture. Write a computer program to plot the number of steps it takes for each number to fall into a $$4 \rightarrow 2 \rightarrow 1$$ cycle. Any visual structures?

b)    Does the mathematics of a $$2 \leftarrow 3$$  connect with Collatz’ problem in some direct way? Ponder and explore. (Both involve playing with products of $$\frac{3}{2}$$ and there is a general feeling that the mathematics needed to answer our questions about the $$2 \leftarrow 3$$ machine will shed light on the Collatz conjecture, and vice versa.)

READING: Have a look Terry Tao’s piece on the Collatz Conjecture for more. There is mathematical interest in the powers of two, the powers of three, and the powers of three halves.

 QUESTION 6:  COUNTING EXPLOSIONS The following table shows the total number of explosions that occur to obtain each representation of the first forty numbers:   Any patterns?

 QUESTION 7:  RATIONAL DECIMAL EXPANSIONS  We can go to “decimals” in a $$2 \leftarrow 3$$ machine.   Here is what $$\dfrac{1}{2}$$ looks like as a decimal in this base. (Work out the division $$1 \div 2$$.) (Do you have choices to make along the way? Is this representation unique?) Some more decimal places:   Can $$\frac{1}{2}$$ have a repeating “decimal” representation in a $$2 \leftarrow 3$$ machine?   What’s a “decimal” representation for $$\dfrac{1}{3}$$ in this machine?   Develop a general theory about which numbers have repeating “decimal” representations in a $$2 \leftarrow 3$$ machine. (I don’t personally have one!)

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