Number treasury 3 investigations facts and conjectures about more than 100 number families 3rd edition

Third Edition

Number Treasury

3

Investigations, Facts and Conjectures
about More than 100 Number Families
Margaret J Kenney • Stanley J Bezuszka
Boston College, Massachusetts, USA

World Scientific


Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Kenney, Margaret J.
Number treasury 3 : investigations, facts, and conjectures about more than 100 number families / by
Margaret J. Kenney (Boston College, USA) and Stanley J. Bezuszka (Boston College, USA). -- 3rd edition.
pages cm
Includes bibliographical references and index.
ISBN 978-9814603683 (hardcover : alk. paper) -- ISBN 978-9814603690 (softcover : alk. paper)
1. Numeration. 2. Mathematical recreations. I. Bezuszka, Stanley J., 1914–2008. II. Title.
III. Title: Number treasury three.
QA141.K46 2015
513.5--dc23

2014040942
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

Copyright © 2015 by World Scientific Publishing Co. Pte. Ltd.
Printed in Singapore


Contents

Foreword

xi

1. A Perfect Number of Investigations 28 = 1 + 2 + 4 + 7 + 14

1

2.

Numbers Based on Divisors and Proper Divisors
Positive Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Divisors, Multiples and Proper Divisors . . . . . . . . . . . . . . . .
Prime and Composite Numbers . . . . . . . . . . . . . . . . . . . .
Sieve of Eratosthenes . . . . . . . . . . . . . . . . . . . . . . . . .
Prime Factorization Property (Fundamental Theorem of Arithmetic) .
Testing for Primes . . . . . . . . . . . . . . . . . . . . . . . . . . .
Divisors of an Integer, GCD, and LCM . . . . . . . . . . . . . . . .
Relatively Prime and Euler φ Numbers . . . . . . . . . . . . . . . .
Abundant, Deficient, and Perfect Numbers . . . . . . . . . . . . . .

Sums and Differences of Abundant and Deficient Numbers . . . . .
Products of Abundant and Deficient Numbers . . . . . . . . . . . .
Multiples of Perfect Numbers . . . . . . . . . . . . . . . . . . . . .
Consecutive Integers and Abundant Numbers . . . . . . . . . . . . .
Abundant Numbers as Sums of Abundant Numbers . . . . . . . . .
Powers of Primes and Deficient Numbers . . . . . . . . . . . . . . .
Even and Odd Integers, Even Perfect Numbers, Mersenne Primes . .
Multiply Perfect Numbers . . . . . . . . . . . . . . . . . . . . . . .
Almost Perfect Numbers . . . . . . . . . . . . . . . . . . . . . . . .
Semiperfect Numbers . . . . . . . . . . . . . . . . . . . . . . . . .
Weird Abundant Numbers . . . . . . . . . . . . . . . . . . . . . . .
Operations on Semiperfect Numbers . . . . . . . . . . . . . . . . .
Primitive Semiperfect Numbers . . . . . . . . . . . . . . . . . . . .
Amicable Numbers . . . . . . . . . . . . . . . . . . . . . . . . . .
Imperfectly Amicable Numbers . . . . . . . . . . . . . . . . . . . .
Sociable Numbers and Crowds . . . . . . . . . . . . . . . . . . . .

39
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39
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42
44
45
46
48
49
53
54
55
56
56
58
61
62
62
63
64
64
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page viii

Practical Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Baselike Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.

Plane Figurate Numbers

73

Polygons . . . . . . . . . . . . . . . . . . . . . . . . . .
Figurate Numbers . . . . . . . . . . . . . . . . . . . . .
Triangular Numbers . . . . . . . . . . . . . . . . . . . .
Operations on Triangular Numbers . . . . . . . . . . . .
Perfect Numbers, Triangular Numbers and Sums of Cubes
Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . .
Triangle Inequality Numbers . . . . . . . . . . . . . . .
Rectangular Numbers . . . . . . . . . . . . . . . . . . .
Square Numbers . . . . . . . . . . . . . . . . . . . . . .
Sums of Square Numbers . . . . . . . . . . . . . . . . .
Positive Square Pair Numbers . . . . . . . . . . . . . . .
Bigrade Numbers . . . . . . . . . . . . . . . . . . . . .
Pythagorean Triples . . . . . . . . . . . . . . . . . . . .
Primitive Pythagorean Triples . . . . . . . . . . . . . . .
Congruent Numbers . . . . . . . . . . . . . . . . . . . .

Fermat’s Last Theorem . . . . . . . . . . . . . . . . . .
Happy Numbers . . . . . . . . . . . . . . . . . . . . . .
Operations on Happy Numbers . . . . . . . . . . . . . .
Happy Number Words . . . . . . . . . . . . . . . . . . .
Repeating Cycles . . . . . . . . . . . . . . . . . . . . .
Patterns in Squares of 1, 11, 111, . . . . . . . . . . . . . .
Squarefree Numbers . . . . . . . . . . . . . . . . . . . .
Tetragonal Numbers . . . . . . . . . . . . . . . . . . . .
Pentagonal Numbers . . . . . . . . . . . . . . . . . . . .
Hexagonal Numbers . . . . . . . . . . . . . . . . . . . .
Recursion and Figurate Numbers . . . . . . . . . . . . .
Remainder Patterns in Figurate Numbers . . . . . . . . .
Gnomic Numbers . . . . . . . . . . . . . . . . . . . . .
Lo-Shu Magic Square; Male and Female Numbers . . . .
4.

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Solid Figurate Numbers
Polyhedra and Solid Figurate Numbers . . . . . . . .
Pyramidal Numbers . . . . . . . . . . . . . . . . . .
Tetrahedral Numbers, Triangular Pyramidal Numbers
Square Pyramidal Numbers . . . . . . . . . . . . . .
Pentagonal Pyramidal Numbers . . . . . . . . . . . .
Hexagonal Pyramidal Numbers . . . . . . . . . . . .
Heptagonal and Octagonal Pyramidal Numbers . . . .

73
74
74
77
78

79
80
83
85
88
89
90
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92
93
94
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95
96
97
98
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100
101
103
107
109
109
111
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113
115
115
118
120
120
121


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Star Numbers and Star Pyramidal Numbers . . . . . . . . .

Rectangular Pyramidal Numbers . . . . . . . . . . . . . .
Cubic Numbers . . . . . . . . . . . . . . . . . . . . . . .
Integers as Sums and Differences of Cubic Numbers, 1729
Pythagorean Parallelepiped Numbers . . . . . . . . . . . .
5.

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More Prime Connections
Goldbach’s Conjectures . . . . . . . . . . . . . . . . .
Integers as Sums of Odd Integers . . . . . . . . . . . .
Integers as Sums of Two Composite Numbers . . . . .
Positive Prime Pair Numbers . . . . . . . . . . . . . .
Prime Line and Prime Circle Numbers . . . . . . . . .
Beprisque Numbers . . . . . . . . . . . . . . . . . . .
A Primes-Between Property . . . . . . . . . . . . . . .
Germain Primes . . . . . . . . . . . . . . . . . . . . .
Twin Primes . . . . . . . . . . . . . . . . . . . . . . .
Semiprimes and Boolean Integers . . . . . . . . . . . .
Snowball Primes . . . . . . . . . . . . . . . . . . . . .
Lucky Numbers . . . . . . . . . . . . . . . . . . . . .
Prime and Lucky Numbers . . . . . . . . . . . . . . .
Polya’s Conjecture about Odd- and Even-Type Integers
Balanced Numbers . . . . . . . . . . . . . . . . . . . .
Fermat Numbers . . . . . . . . . . . . . . . . . . . . .
Cullen Numbers . . . . . . . . . . . . . . . . . . . . .
Ruth–Aaron Numbers . . . . . . . . . . . . . . . . . .

6.

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page ix

121
122
125
126
128
131

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Digital Patterns and Noteworthy Numbers
Monodigit and Repunit Numbers and Langford Sequences
Social and Lonely Numbers . . . . . . . . . . . . . . . .
Additive Multidigital Numbers . . . . . . . . . . . . . .
Multiplicative Multidigital Numbers . . . . . . . . . . .

Kaprekar’s Number 6174, 99 and 1089 . . . . . . . . . .
Doubling Numbers . . . . . . . . . . . . . . . . . . . . .
Good Numbers . . . . . . . . . . . . . . . . . . . . . . .
Nearly Good Semiperfect Numbers . . . . . . . . . . . .
Powerful Numbers . . . . . . . . . . . . . . . . . . . . .
Armstrong Numbers and Digital Invariant Numbers . . .
Narcissistic Numbers . . . . . . . . . . . . . . . . . . .
Additive Digital Root Numbers . . . . . . . . . . . . . .
Additive Persistence of Integers . . . . . . . . . . . . . .
Multiplicative Digital Root Numbers . . . . . . . . . . .
Multiplicative Persistence of Integers . . . . . . . . . . .

131
133
134
134
136
138
138
139
140
140
142
142
144
145
147
147
148
149

151

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151
153

155
156
157
159
161
163
164
165
167
167
169
170
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page x

Modest and Extremely Modest Numbers . . . . . . . . . . . . . . . . . . . . 173
Visible Factor Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Nude Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.

More Patterns and Other Interesting Numbers
More Sums of Consecutive Integers . . . . . . . . . . . . . . . .
Product Patterns for Consecutive Integers . . . . . . . . . . . . .
Consecutive Integer Divisors . . . . . . . . . . . . . . . . . . .
Consecutive Number Sums and Square Numbers . . . . . . . . .
Factorial Numbers, Applications and Extensions . . . . . . . . .
Factorial Sum Numbers and Subfactorial Numbers . . . . . . . .
Hailstone and Ulam Numbers; The Collatz and Ulam Conjecture
Palindromic Numbers . . . . . . . . . . . . . . . . . . . . . . .
Creating Palindromic Numbers . . . . . . . . . . . . . . . . . .
Palindromic Number Words and Curiosities . . . . . . . . . . .
Palindromic Numbers and Figurate Numbers . . . . . . . . . . .
Palindromic Primes and Emirps . . . . . . . . . . . . . . . . . .
Honest Numbers . . . . . . . . . . . . . . . . . . . . . . . . . .
Bell Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . .
Catalan Numbers . . . . . . . . . . . . . . . . . . . . . . . . .
Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . .
Lucas Numbers . . . . . . . . . . . . . . . . . . . . . . . . . .
Tribonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . .
Tetranacci Numbers . . . . . . . . . . . . . . . . . . . . . . . .
Phibonacci Numbers . . . . . . . . . . . . . . . . . . . . . . . .
Survivor Numbers or U-Numbers or Ulam Numbers . . . . . . .
Tautonymic Numbers . . . . . . . . . . . . . . . . . . . . . . .
Lagado Numbers . . . . . . . . . . . . . . . . . . . . . . . . . .

177
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177
181
183
184
185
188
189
191
193
194
196
196
197
198

200
203
206
207
208
209
209
210
211

Recommended Readings

215

Glossary of Numbers

217

Solutions to Investigations

225

Solutions to Exercises

249

Index

305

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Foreword

The gift of numbers like the gift of fire has made the world much brighter.
Stanley J. Bezuszka (1914–2008)

Introduction
Time and numbers were born together. Time is a measure of change, and numbers
express that measure. There is an awesome mystery that surrounds the all-pervasive
dimension of time. Likewise, there is an awesome mystery and fascination about
numbers that attract distinguished mathematics researchers as well as imaginative
amateurs the world over. Teachers who pursue numbers and the rich history of
numbers with their classes can provide students with an understanding that mathematics is a collaborative effort that has been nurtured by individuals and groups
representing many cultures and periods of history. Students can learn to become
accomplished investigators, to make discoveries, and contribute to a branch of
mathematics that is vibrant and motivating. Number Treasury3 has evolved in
order to serve as a catalyst for those who ascribe to this point of view.

Details
Number Treasury3 is a broadening and update of Number Treasury2 . The book

contains information about more than 100 families of positive integers. Brief historical notes often accompany the descriptions and examples of the number families.
Exercises for each major family are provided to stimulate insight. Some exercises
contain problems that are thought provokers to be resolved simply with paper and
pencil; others should be tackled with calculator in hand so that lengthier computations can be managed with ease and take the results to a higher level of understanding. Still other problems are intended for more extensive exploration with the
use of computer software. In some instances it is helpful to model problems with
hands-on materials.

page xi


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The emphasis in Number Treasury3 is on doing rather than proving. However, the
reader is urged to think critically about situations, to provide reasoned explanations,
to make generalizations and to formulate conjectures. The book begins with a
chapter of Investigations. These are principally stand-alone activities that represent
content drawn from the Chapters 2 through 7 of the book. Their purpose is to set the
tone of the book and to stimulate student reflection and research in a variety of areas.
In fact, throughout the book, the reader will find numerous open-ended problems.
This book also contains detailed solutions to the Exercises and Investigations.
A Glossary and Index are provided for quick access to information. References
and recommended readings are supplied so that teachers and students can use this

book as a stepping stone to more concentrated study.

Who Uses Number Treasury3
This book is written for teachers and students. For teachers Number Treasury3
is a resource for instructional preparation and problems, together with snapshots
of mathematical history intended for teachable moments. For students who are
engaged in learning about number families and who are assigned problems, projects
and papers, Number Treasury3 is a useful source of ideas and topics. The mix of
discussion with examples and illustrations is intended to serve as a writing model
for the student. Both audiences should think critically about the content, provide
carefully reasoned explanations, make generalizations, and form conjectures.

Who Is Involved
The first edition was completed with the able assistance of six Boston College
graduates and undergraduates: Jeanne Cavanaugh, James Cavanaugh, Claudia
Katze, Stephen Kokoska, Jill Nille, and Jonathan Smith.
Seven Boston College graduates and undergraduates were indispensable in the
production of the second edition. Special thanks and grateful appreciation go to
Joan Martin for her thoughtful content and style suggestions, editorial advice and
word processing skills; to Cynthia Tahlmore, Geraldine Mele, and Erin Mitchell
for computer graphics and word processing assistance; to Allyson Russo, Shannon
Toomey, and Megan Mazzara for problem solutions.
The third edition has been completed by the surviving original author with the
invaluable assistance and perseverance of Geraldine Mele who offered not only
content suggestions but who also especially contributed word processing, computer
graphics and style expertise. Sincere gratitude and appreciation is also extended to
Joan Martin for her careful review of the manuscript.

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Chapter 1

A Perfect Number of Investigations
28 = 1 + 2 + 4 + 7 + 14
A GREAT discovery solves a great problem but there is a grain of discovery in any problem.
George Polya (1887–1985)

What are They?
The Investigations that follow are a set of stand-alone activities. Each Investigation
focuses on at least one number family or topic relating to numbers. All but six of
the Investigations are described on one page. Share with students that the problems
in an Investigation are intended to be challenging in many ways:
• The time needed to complete an Investigation may vary and exceed the time
required to finish a typical homework assignment.
• The computation necessary to bring closure may be lengthy and demanding —
even with the use of technology.
• The amount of writing, discussing, explaining and illustrating may be more than
anticipated.

Teacher Tips
The Investigations are listed in ascending order of difficulty in the table on the next
two pages. There are three levels of difficulty represented in the 28 Investigations
that can be assigned individually or adapted for group work. The lowest level
consists of the first eight Investigations that have the least prerequisites. The middle
level consists of the next nine Investigations and requires more use of abstract
reasoning and familiarity with algebraic expressions. The final 11 Investigations
challenge the student to persist and probe more deeply in order to complete the
work.
There is also a column in the table naming the most significant prerequisite(s)
needed by the student to understand and carry out the work in each Investigation.
The teacher may choose to provide additional content background for some specific Investigations prior to assigning them. Assign the Investigations as extended
homework or as in-class work. Some Investigations call for the preparation of

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reports. Thus, students may need further directions, especially about the kind of
resources available for them to use. Students should know the Internet is an excellent resource, and that it should be used appropriately as they compile their reports.
Finally, pages noted in the Prerequisites column refer to related material contained
in Chapters 2 through 7.

Page

Investigation

4
5,6
7

Footsteps of Lagrange
Trying Trapezoids
Hexagons in Black & White

8,9

Marble Art

10

Honest Number Hunt

11

Seeking Honesty in Numbers

12

Geoboard Journeys

13

Mysterious Mountains & Binary
Trees
Fermat Factorings
Factor Lattices
A Juggling Act
The Super Sum
Conjecturing with Pascal
Pythagorean Triple Pursuits
Pentagonal Play
Triangular Number Turnarounds
Centered Triangular Numbers

14
15,16
17
18
19,20
21
22
23
24
25,26
27
28,29

30
31

Catalan Capers
Highly Composite Numbers
Tower of Hanoi & the Reve’s
Puzzle
Perfect Number Patterns
Crisscross Cubes

Prerequisites & Text reference
Square numbers, p. 85
Triangular numbers & trapezoids, p. 74
Familiarity with recursive & explicit
formulas, p. 107
Figurate numbers,
following directions, pp. 74–111
Counting in a language, searching
resources, p. 197
Groups work together to organize their
data, p. 197
Trial/Error pursuit,
link Catalan & Pascal, p. 200
Doing & arranging sketches, p. 200
Prime factorization, p. 42
LCM, prime factorization, p. 46
Describing systematically, p. 46
Reasoning with patterns, p. 74
Articulating patterns, p. 79
Evaluating expressions, p. 90

Squares & triangles within, p. 101
Visualizing triangles, p. 80
Making algebraic generalizations,
p. 74
Connect algebra & geometry, p. 201
Counting divisors, p. 45
Recursive actions & thinking, p. 56
Using logs to count, p. 59
Perimeter, area, volume, make
generalizations, p. 125

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Page

Investigation

32

A Medieval Pattern

33
34
35

Prime Magic
Dealing with Digits (base ten)
Factorial Finishes

36
37

Designing Designs
Fibonacci Fascinations

Prerequisites & Text reference
Connect pattern to shape &
number, p. 88
Trial & error yields results, p. 136
Exploring ways to count, p. 188
Importance of 2 × 5 in reasoning,
p. 185
Representation is critical, p. 198
Spreadsheet use, p. 203

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Footsteps of Lagrange
Develop the first 20 terms of the sequence sn , where sn is the number of ways n
can be written as a sum of at most 4 squares. Note one term will be counted as
a sum.
EXAMPLE
How many ways can 5 be written as a sum of at most 4 squares?
1 way since 5 = 4 + 1 = 22 + 12 .
EXAMPLE
How many ways can 9 be written as a sum of at most 4 squares?
2 ways since 9 = 22 + 22 + 12 and 9 = 32 .
1. Fill in the following table.

n

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Ways to Write n as
Sum of at Most 4 Squares

Number
of Ways

2 2 + 12

1

3 2 ; 22 + 2 2 + 1 2

2

2. Describe in a few sentences some patterns you observe in the table.
3. Find a number and verify that it can be written as a sum of at most 4 squares in
exactly
a) three ways
b) four ways

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5

Trying Trapezoids
Trapezoidal numbers are figurate numbers that can be represented by trapezoidal
arrays of two or more rows of dots or chips.
For example, the number 5 can be pictured as a trapezoidal number using two rows
of dots or chips:

•

• •
• •

• •
• • •

Isosceles array

Right-angled array

There are many different ways to describe trapezoidal numbers. Two of these are
shown below.
• Subtracting triangular numbers produces trapezoidal numbers.
EXAMPLE
Start with T4 . Subtract T1 and T2 .
T4 − T1

T4 − T2

9

7

Thus 7 and 9 are trapezoidal numbers.
1. Using subtraction of triangular numbers procedure, name the trapezoidal numbers that come from
a) T5

b) T8

c) Tk , k > 2

• Adding groups of consecutive numbers greater than 1 produces trapezoidal
numbers.
EXAMPLE
Trapezoidal numbers that consist of 3 rows, one of which has 4 dots, are

9

12

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2. Name the trapezoidal numbers that consist of
a) 4 rows one of which has 3 dots.
b) 5 rows one of which has 4 dots.
c) 3 rows one of which has k dots, k > 1.
3. List all ways that each given number can be trapezoidal.
a) 11
b) 27
c) 33
4. The trapezoidal number 9 can be pictured in isosceles or right-angled form:

isosceles

right-angled

Study the right-angled model and develop a formula for a trapezoidal number
in terms of the number of rows and the first and last rows.
5. When expressed in right-angled form it is clear that a trapezoidal number in the
sequence 2 + 3, 2 + 3 + 4, 2 + 3 + 4 + 5, . . . can be written as the sum of a
2 × n rectangular number and a triangular number.
For example,

2+3+4 = 6+3

Write each number as the sum of a 2 × n rectangular number plus a triangular
number.

a) 2 + 3 + 4 + 5
c) 2 + 3 + 4 + · · · + 10
b) 2 + 3 + 4 + 5 + 6
d) 2 + 3 + 4 + · · · + 20
6. Segments joining the dots in the trapezoidal number 2 + 3 + 4 = 9 form an
array
1 hexagon with 2 dots per side and 2 triangles.

Determine that segments joining the arrays of dots
a) 2 + 3 + 4 + 5 + 6 forms 3 hexagons and 6 triangles
b) 2 + 3 + 4 + 5 + 6 + 7 + 8 forms 6 hexagons and 12 triangles.
Predict, then verify, the number of hexagons and triangles for the arrays of dots
c) 2 + 3 + · · · + 10
d) 2 + 3 + · · · + 12
e) 2 + 3 + · · · + 20

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7

Hexagons in Black and White
1. To produce the patterns below you will need a quantity of two different colored
chips. Copy each of the designs shown and then create the next two that follow
the pattern. Use the designs to produce the data in the table. Let n be the number
of hexagon paths around the center chip.
Number
of Black
Chips

Number
of White
Chips

Total
Chips

C
C
C
C

A recursive formula for a sequence defines each term of the sequence using the
preceding term or terms.
An explicit formula for a sequence defines each term of the sequence by a rule
that depends only on the term number.

2. a) Write a recursive relation for the total number of chips Cn in row n of the
table. That is, express Cn in terms of Cn−1 .
b) Give an explict formula f(n) for the total number of chips in terms of n.
3. Create your own patterns using two different colored chips. Gather the data and
record it. Summarize your findings with a recursive and an explicit formula.


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Marble Art
Polyhedral numbers can be represented physically in a variety of ways. For
example, you can produce effective and attractive models from practice golf
balls, holiday glass ball ornaments, or marbles held together with glue. Two
models using marbles are described below. Try these and then create some
variations of your own.

Square pyramidal numbers
1. Make glued layers of marbles representing the consecutive square numbers. Let

them stand until they are completely dry.

2. Start with the 5 × 5 layer. Place and glue the 4 × 4 layer on top so that it rests
in the indentations of the 5 × 5 layer. Continue by placing and gluing the 3 × 3
layer, the 2 ×2 layer, and so on. When all layers are glued in place, let the model
stand until dry.
3. For a better effect you can glue 4 single marbles at each of the corners to serve
as feet so that the model is raised from the surface. A mirror can be positioned
under the model to provide interesting reflections.

Tetrahedral cluster
1. Build and glue a hexagonal array that has 3 marbles per side. Let it stand until
completely dry.

2. Create 6 feet for the hexagonal array, 3 feet each for the top and bottom sides
of the surface. Each foot here is composed of 4 marbles (1 marble glued onto
the indentation of a triangular array of 3 marbles).
Foot

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9

3. Apply glue and attach the three feet to each side of the hexagonal layer as shown
in the figure. Attach one additional marble where the 3 feet meet in the center
on each side of the hexagonal array as shown in the diagram. Let the model
stand until completely dry.

Top and bottom view


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Honest Number Hunt
An honest number in a language is a number whose word letter count and size
are equal.
In English, “four” represents as many objects as there are letters in the word four.
In fact, four is the only honest number in English. In Spanish, "cinco" represents
as many objects as there are letters in the word cinco. Cinco is an honest number
in Spanish.
1. Undertake an extensive search for honest numbers in as many languages as
possible. Prepare a chart with your class that displays the honest numbers found.
Do all the languages you examined have at least one honest number?
2. Honest numbers in a language have an interesting property as the following
example illustrates.
English

EXAMPLE

Choose a number.
Write the number word for 2020.
Count and record the number of letters.
Write the number word for 17.
Count and record the number of letters.
Write the number word for 9.
Count and record the number of letters.
Write the number word for 4.

2020
two thousand twenty
17
seventeen
9

nine
4
four

If you continue, 4 and four keep repeating.
The number trail has 4 numbers and ends in the honest number 4.
2020

17

9

4

a) Using the English language, apply this process to other numbers. Summarize
your findings.
b) Choose another language in which you know how to count and which has an
honest number. Apply the process described in the example to several numbers
in this language. In a paragraph or two, summarize your findings for a class
report.

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11

Seeking Honesty in Numbers
An honest number in a language is a number whose word letter count and size
are equal.
Ten represents 10 objects, but 10 has 3 letters. Ten is not an honest number.
Four represents 4 objects and four has 4 letters. Four is an honest number.
In the English language, 4 is the only honest number.
Determine which number trails end in the honest number 4.
EXAMPLE
Choose a number.
Write the number word for 1066.
Count and record the number of letters.
Write the number word for 19.
Count and record the number of letters.
Write the number word for 8.
Count and record the number of letters.
Write the number word for 5.
Count and record the number of letters.

1066
one thousand sixty-six

19
nineteen
8
eight
5
five
4

If the steps are continued, 4 and four keep repeating.
This number trail has five numbers and ends in 4.
1066

19

8

5

4

Find the number trails for:
1. 63
2. 163
3. 999
4. 8999

5. any five-digit number
6. any six-digit number
7. any seven-digit number
8. any eight-digit number

With your class, produce a table or diagram that shows all possible number trails
for the first 200 integers.


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Geoboard Journeys
Use a geoboard of size 5 pegs by 5 pegs or larger, together with a sheet of square
dot paper for recording results.
Your task: Count the number of paths from A to B that are composed of only
horizontal or vertical segments. You may move only to the right or up. Remember
to use only pegs that are on or below the diagonal from A to B.
EXAMPLES
On a 2 × 2 geoboard there is 1 path.
B
A

On a 3 × 3 geoboard there are 2 paths.
B

A

B

A

In problems 1 and 2, A is the lower left peg and B is the upper right peg.
1. Produce the acceptable paths from A to B on a
a) 4 × 4 geoboard
b) 5 × 5 geoboard
and record the results on dot paper. Try to be systematic in recording solutions.
2. Based on the solutions for the 2 × 2, 3 × 3, 4 × 4, and 5 × 5 sizes, identify if you
are able the kind of numbers that represent the total path count from A to B.
3. Using only horizontal or vertical segments and any pegs for B, count all acceptable paths from A to B. Give the total and describe your counting strategy on a
a) 2 × 2 geoboard
c) 4 × 4 geoboard

b) 3 × 3 geoboard
d) 5 × 5 geoboard

4. Can you discover a connection between your solutions to problems 2 and 3 and
the numbers in Pascal’s triangle? Explain your thinking.
5. Create and solve your own geoboard journey problem.

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13

Mysterious Mountains and Binary Trees
Mountains are constructed with up-segments and down-segments.

1 mountain design is
possible using one up–
one down-segment.

2 mountain designs are possible using two
up–two down-segments.

1. Draw all mountain designs composed of 3 up-segments and 3 down-segments.
2. Draw all mountain designs composed of 4 up-segments and 4 down-segments.
A rooted binary tree starts from a vertical edge. Rooted binary trees will be ordered
by the number of interior vertices. One binary tree is different from another if it

cannot be turned to match the other.
Here are some rooted binary trees.

3. Draw all different rooted binary trees with 3 interior vertices.
4. Draw all different rooted binary trees with 4 interior vertices.
5. Describe any connection you can make between the mountains and the trees.


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Fermat Factorings
Pierre de Fermat (1601–1665) a French lawyer by profession and an amateur
mathematician by choice developed an algorithm for finding the prime factorization
of an integer. Fermat’s algorithm is applied to odd integers that are not squares.
His strategy consisted in adding consecutive odd integers to N until a square was
reached. If the given number is even, first divide by 2 until an odd non-square
integer N appears.
EXAMPLE

Find the prime factorization of 105.
105
+1
106
+3
109
+5
114
+7
121 = 112

Since 121 is a square, then 105 can be written as
(11 + − )(11 − − ), where − is replaced by the
number of odd integers needed to reach a perfect
square. In this case, 4 odd integers are added.
Thus 105 = (11 + 4)(11 − 4) = 15 × 7.
Now factor 15 by this procedure.
15
+1
16 = 42
So 15 = (4 + − )(4 − − ) = (4 + 1)(4 − 1) = 5 × 3
Finally, 105 = 3 × 5 × 7.

EXAMPLE
Find the prime factorization of 84.
Since 84 is even, divide by 2 until an odd non-square integer appears.
21
+ 1
22
+ 3

25 = 52

84 = 2 × 42 = 2 × 2 × 21
Then 21 = (5 + − )(5 − − )
= (5 + 2)(5 − 2)
=7×3
Finally,
84 = 22 × 3 × 7.

1. Use Fermat’s algorithm to find the prime factorization of
a) 185

b) 360

c) 237

d) 500

2. Explain why Fermat’s algorithm works.
3. Write a report on the life of Pierre de Fermat, including names of his mathematical friends and some of the mathematical problems he worked on. Be sure to
include information about Fermat primes, Fermat’s Last Theorem, and Andrew
Wiles.

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A Perfect Number of Investigations 28 = 1 + 2 + 4 + 7 + 14

15

Factor Lattices
The collection of divisors of an integer can be represented using vertices and line
segments, in linear, two, three, and higher dimensional arrays. Such figures are
called factor lattices and each vertex of these figures is labeled with one of the
divisors of the integer. Factor lattices can be produced using prime factorization
and the concept of least common multiple.
The factor lattice for any number that is a power of a prime is linear. For example,
the factor lattice for 81 = 34 is:
0

The factor lattice for 81 has 5 vertices, one for each divisor of 81. Also, the factor
lattice has four unit segments.
Observe that powers of the prime 3 are matched in order starting with 30 . Check
to see that the factor lattice for 27 has eight vertices and seven unit segments.
1. The factor lattices for 75 and 36 are shown below. Note that lcm is an abbreviation for least common multiple. Study them carefully for clues.
prime

prime

2

lcm(3,2)

2

lcm(6,4)

prime

2

prime

2

lcm(3,5)

lcm(15,5 )

lcm(9,6)
2

75 = 3 • 5 2

36 = 2 • 3

lcm(18,12)
2

Use them as guides and draw factor lattices for:
a) 24

b) 72

c) 100

Numbers whose prime factorization is a single prime to some power have a factor
lattice that is one dimensional, while numbers whose factorization consists of


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two distinct primes have factor lattices that are two dimensional. It appears that
a number that has three distinct prime factors should have a three dimensional
factor lattice. Indeed, the factor lattice for 30 can be represented by the vertices
and edges of a cube.

2 prime

1

3
prime
10
lcm(2,5)

5
prime

6 lcm(2,3)

30
lcm(2,3,5)

15
lcm(3,5)

2. Name 5 other numbers whose factor lattice is also a cube. Using the example
for 30, draw the cubes and match divisors with vertices.
Try building models of three dimensional factor lattices. You can use toothpicks as
edges with gum drops or mini-marshmallows for vertices or commercially available
materials.
3. Name a number whose factor lattice is the figure shown below.

1

Check your guess by assigning divisors to vertices.

4. Draw and label the factor lattices for 90 and 168.
5. Match the vertices of a four dimensional hypercube with the divisors of the
number 210.

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17

A Juggling Act
You have two unmarked containers that you take to a well.

a

b

One container holds a liters, the other b liters of water. Your job is to obtain exactly
1 liter of water in one of the containers. Explain in detail how you will accomplish
this if:
1. a = 5 b = 7
2. a = 4

b=7

3. a = 5

b=8

4. Describe in detail how you could use a 5-liter container and an 8-liter container
to get each of the amounts 1, 2, 3, . . . ,13 liters.
5. If you have a 6-liter and an 8-liter container, can you obtain exactly 1 liter of
water? Explain your answer.
6. When a and b are given, how can you tell whether it is possible to obtain exactly
1 liter of water?
7. You have three containers whose capacities are 8, 5, and 3 liters.
The 8-liter container is full of water. Describe how to split this water you have
into two equal amounts.

8 liters

5 liters

3 liters