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Adventures in Exploding Dots

LESSON 9

Teacher Guides brought to you by THE GLOBAL MATH PROJECT

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The Global Math Project was founded in 2016 with the mission of showing that learning mathematics can and should inspire joy. Over 6 million students and teachers across the planet have looked at our curriculum materials and used them to ignite and sustain their love of mathematics.

More than 95% of teachers agree that our lessons make mathematics more approachable, enjoyable, and understandable.

Our mathematics all starts with the story of Exploding Dots, which is the story of place-value: how we write and do arithmetic with numbers, and how the joy of algebra and much more opens from there.

Here we present six lessons for ages 10 and up that get the young—and the young at heart—started on this wonderful journey.

The full Exploding Dots experience, leading into high-school mathematics, college mathematics, and unsolved research challenges, can be found at www.globalmathproject.org. (All so naturally and readily follows from what we present here!)

Enjoy!

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Contents:

Overview of Lesson 9 … 4

Lesson 9 Going Completely Wild: Base One-and-a-Half … 5 Let's go to the wild side and play with the mysteries of a fractional base!

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Lesson 9 OVERVIEW Going Completely Wild:

Base One-and-a-Half Let's go to the wild side and play with the mysteries of a fractional base.

OBJECTIVES

The beauty of the set-up of an Exploding Dots machine is that it is immediately amenable to tweaking and contorting and so allows for the discovery of non-traditional bases. This opens up a new universe of mathematical intrigue and delight—and mystery! There are many unsolved questions about numbers in these unconventional bases.

The Experience in a Nutshell

Consider a 2 3← machine for which three dots in any one box “explode” to be replaced with two dots one place to their left. For example, placing ten dots into the machine yields the code 2101 .

This is the number ten represented in base one-and-a-half!

Let’s explore the mysteries of these codes.

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Lesson 9 Teacher Guide

Going Completely Wild: Base One-and-a-Half

Let's go to the wild side and play with the mysteries of a fractional base.

PART 1: Baes One-and-a-Half … 7

PART 2: Divisibility Rules … 14

PART 3: Counting Digits … 18

PART 4: Palindromes … 20

PART 5: Closing … 21

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Further Resources for this Lesson Videos: Exploration 9 of Exploding Dots at www.globalmathproject.org.

Web App: Interactive online play at www.explodingdots.org. Collect KAPOWS!

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Part 1: BASE ONE-AND-A-HALF Let’s get weird!

What do you think of a 1 1← machine?

What happens if you put in a single dot? Is a 1 1← machine interesting? Helpful?

What do you think of a 2 1← machine?

What happens if you put in a single dot?

What do you think of the utility of a 2 1← machine?

After pondering on these machines for a moment you might agree there is not much one can say about them. Both fire “off to infinity” with the placement of a single dot and there is little control to be had over the situation.

How about this then?

What do you think of a 2 3← machine?

This machine replaces three dots in any one box with two dots one place to their left.

Ah! Now we’re on to something. This machine seems to do interesting things.

For example, placing ten dots into the machine

first yields three explosions,

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then another two explosions,

followed by one more.

We see the code 2101 appear for the number ten in this 2 3← machine.

Show that the number thirteen has code 2121 .

Here are the 2 3← codes for the first fifteen numbers. (Check these!)

Does it make sense that only the digits 0 , 1, and 2 appear in these codes?

Does it make sense that the final digits of these codes cycle 1,2,0,1,2,0,1,2,0,...?

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Even if we don’t know what these codes mean, we can still do arithmetic in this weird system! For example, ordinary arithmetic says that 6 7 13+ = , and the codes in this machine say the same thing too.

Six has code 210 . Seven as code 211 . Adding gives 2121 , which is the code for thirteen.

But the real question is:

What are these codes? Are these codes for place-value in some base? If so, which base?

Of course, the title of this lesson gives the answer away, but let’s reason our way through the mathematics of this machine, nonetheless.

Dots in the rightmost box, as always, are each worth 1. Let’s call the values of dots in the remaining boxes x , y , z , w , … .

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Now three dots in the 1s place are equivalent to two dots in the x place.

This tells us that 2 3 1x = ⋅ , giving the value of 32

x = , one-and-a-half.

In the same was we see that 32 32

y = ⋅ .

This gives23 3 3

2 2 2y = ⋅ =

, which is 9

4.

And we also see,

232 32

z =

giving 33

2z =

, which is 27

8,

332 32

w =

giving 43

2w =

, which is 81

16,

and so on.

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We are indeed working in something that looks like base one-and-a-half!

Comment: Members of the mathematics community might prefer not to call this base-one-a-half in a technical sense since we are using the digit “2” in our work here. This is larger than the base number. To see the language and the work currently being done along these lines, look up beta expansions and non-integer representations on the internet. In the meantime, understand that when I refer with “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 mathematics that arises from this particular 2 3← machine.

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I personally find this version of base one-and-a-half intuitively alarming! We are saying that

each and every integer can be represented as a combination of the fractions 1, 32

, 94

, 278

, 8116

,

and so on. These are ghastly fractions!

For example, we saw that the number ten has the code 2101 .

Is it true that this combination of fractions, 27 9 32 1 0 1 18 4 2

× + × + × + × , turns out to be the

perfect whole number ten?

Yes!

And to that, I say: whoa!

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There are plenty of questions to be asked about numbers in this 2 3← machine version of base one-and-a-half, and many represent unsolved research issues of today. For reference, here are the codes to the first forty numbers in a 2 3← machine (along with zero at the beginning).

Over the next few pages, I present some specific ideas to possibly mull on. Some of the questions I pose are open problems, still baffling mathematicians to this day!

Does it make sense that all the codes, after a small initial “hiccup” begin with the digit 2 ?

Does it make sense that, after a bigger initial “hiccup,” all the codes begin with 21?

After an even bigger hiccup, do the first three digits of the codes stabilize?

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Part 2: DIVISIBILITY RULES

Divisibility by Three

Look at the list of the first forty 2 3← codes of numbers. As soon as one realizes why the final digit of these codes cycle through the values 0, 1 and 2, the following divisibility rule for 3 makes sense.

A number written in 2 3← code is divisible by three precisely when its final digit is zero.

Find, and explain, a divisibility rule for 9.

Find divisibility rules for other powers of three.

Divisibility by Five

Here is something curious.

A number is divisible by 5 only if the alternating sum of the digits of its 2 3← machine code is a multiple of five.

For example,

Twenty has code 21202 and 2 1 2 0 2 5− + − + = is a multiple of five.

Forty has code 2101121 and 2 1 0 1 1 2 1 0− + − + − + = is a multiple of five.

Eleven has code 2102 and 2 1 0 2 1− + = = − is not a multiple of five.

To explain this, we have three matters to consider:

• Things being multiple of fives • Alternating sums of digits • The mechanics of a 2 3← machine

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To bring these three ideas together, let’s see what an explosion in a 2 3← machine does to an alternating sum of digits with regard to multiples of five. (Got that?)

In an alternative sum we’ll be considering

a b− + changing to ( ) ( )2 3 5a b a b− + + − = − + − or a b− changing to ( ) ( )2 3 5a b a b+ − − = − + . Either way, an explosion in a 2 3← machine does not affect whether or not the alternating sum of digits you have so far is a multiple of five: an explosion just changes that alternating sum up or down by five. So, if we put in N dots in the rightmost box of a 2 3← machine, with alternating sum

0 0 0 N N− + − + = , and perform explosions to get its 2 3← machine code

| | | | |a b c d e , the alternating sum of this code a b c d e− + − − + differs from N by a multiple of five. So N is a multiple of five precisely if the alternating sum is, just as claimed.

Have we just proved that in any m n← machine, a number is divisible by m n+ precisely when the alternating sum of the digits of its code is? (Does this seem to fit the standard divisibility rule for eleven in base ten?)

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Divisibility by Two

What is a divisibility rule for the number two for numbers written in 2 3← code?

What common feature does every second code have?

I personally do not know a swift way to tell whether or not a number is even just by looking at its 2 3← machine code. Nor do I think anyone on this planet knows a simple divisibility rule for the number two! If you find one, let me—and the world—know!

By the way ….

Delete the final digit of the 2 3← machine code of any number and what results is the machine code of an even number.

For instance, forty has code 2101121. Delete the final digit and you get 210112, which is the code for the even number twenty-six. Delete its final digit and you get 21011 (sixteen), then 2101 (ten), then 210 (six), then 21 (four), then 2 (two).

Why does this property hold? Why does deleting the final digit give you two-thirds of the original number rounded down to an integer (and why is this sure to be even)?

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Divisibility by One?

This is a silly title! Every number is divisible by one.

What I really mean to ask here:

How do we know if a given collection of 0 s, 1s, and 2 s is the code of an integer?

For example, if look at the list of the first forty 2 3← machine codes we see that “ 201” is

skipped. This combination of powers of one-and-a-half is not an integer: it’s the number 152

.

Suppose I asked:

Is

2102212020120020122011201102202010221020100202212

the code for a whole number in a 2 3← machine?

Of course, you could just work out the sum of powers this represents and see whether or not that sum is a whole number. But that doesn’t seem fun!

Is there some quick and efficient means to look as a sequence of 0 s, 1s, and 2 s and determine whether or not it corresponds to the code of a whole number? (Of course, how one defines “quick” and “efficient” is up for debate.)

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Part 3: COUNTING DIGITS

Looks again at the first forty 2 3← codes.

Notice

0 gives the first one-digit code. (Some might prefer to say 1 here.)

3 gives the first two-digit code.

6 gives the first three-digit code.

9 gives the first four-digit code.

and so on.

This gives the sequence: 3, 6, 9, 15, 24, ….

(Let’s skip the questionable start.)

Are there any patterns to this sequence?

Can you guess what the next number in the sequence is going to be?

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If you are thinking something Fibonacci-esque ( 3 6 9+ = , 6 9 15+ = , 9 15 24+ = , and so the next number is 15 24 39+ = ), then, sadly, you will be disappointed to learn that the next number in the sequence is 36 .

The sequence is

3, 6, 9, 15, 24, 36, 54, 81, 123, 186, 279, 420, 630,

GOING FROM ONE TERM TO THE NEXT

Given one number in the sequence it is possible to predict what the next term shall be.

If M is an even number in the sequence, then the next number in the sequence is 32

M .

If M is odd, then the next number is ( )3 12

M + .

Can you see this? If M dots are needed in the rightmost box to first make a code k digits long, then we need M dots to appear in the second-to-right most box to first make a code with 1k + digits. So, how many dots do we need to put into the rightmost box so that, after explosions, we have those M dots there?

AN EXPLICIT FORMULA?

What’s the hundredth term of the sequence?

We could work it out by working our way up from the 13th term (630), to the 14th term, to the 15th term, and so on, all the way up to the 99th term, and then the 100th term. But that would be tedious!

What would be better is to have an explicit formula for the n th term of the sequence. Then, to work out the 100th term, all we need to do is plug in 100n = into that formula and be done!

Do you know such a formula? (It seems no one does!)

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Part 4: PALINDROMES The number 17 has 2 3← machine code 21012 , which is a palindrome, it reads the same way forwards as backwards.

The codes for 1, 2, 5, 8, 35, 170, 278, 422, and 494 are also palindromes.

Are there any more examples in base one-and-a-half?

No one knows! (Though Dr. James Propp argues in his essay here that there might not be any more examples.)

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Part 5: CLOSING

We’ve asked here just a few curious, unsolved questions about the powers of one-and-a-half. There are more, many more!

Understanding the powers of this basic fraction is an open area of research.

(For example, see Field Medalist Terry Tao’s piece here. Also see “On Base 3/2 and its Sequences” by ben Chen et al (August, 2018) for additional work on the codes of numbers that arise from a 2 3← machine.)

It is astounding to me that we don’t understand the mathematical properties of such a simple fraction. That’s just wonderful!

If you want to see some more questions about base one-and-a-half and start exploring other strange bases (negative bases, irrational bases), look at Exploration 9 of the Exploding Dots story here.

Mathematics is truly a wonderland of play, intrigue, deep mystery, and deep delight.