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

46 Abundant, Deficient, and Perfect Numbers.. 48 Sums and Differences of Abundant and Deficient Numbers.. 77 Perfect Numbers, Triangular Numbers and Sums of Cubes.. 115 Tetrahedral Numbe

World Scientific

Investigations, Facts and Conjectures about More than 100 Number Families

Margaret J Kenney • Stanley J Bezuszka

Boston College, Massachusetts, USA

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

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

Foreword xi

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

2 Numbers Based on Divisors and Proper Divisors 39

Positive Integers 39

Divisors, Multiples and Proper Divisors 39

Prime and Composite Numbers 40

Sieve of Eratosthenes 41

Prime Factorization Property (Fundamental Theorem of Arithmetic) 42

Testing for Primes 44

Divisors of an Integer, GCD, and LCM 45

Relatively Prime and Eulerφ Numbers 46

Abundant, Deficient, and Perfect Numbers 48

Sums and Differences of Abundant and Deficient Numbers 49

Products of Abundant and Deficient Numbers 53

Multiples of Perfect Numbers 54

Consecutive Integers and Abundant Numbers 55

Abundant Numbers as Sums of Abundant Numbers 56

Powers of Primes and Deficient Numbers 56

Even and Odd Integers, Even Perfect Numbers, Mersenne Primes 58

Multiply Perfect Numbers 61

Almost Perfect Numbers 62

Semiperfect Numbers 62

Weird Abundant Numbers 63

Operations on Semiperfect Numbers 64

Primitive Semiperfect Numbers 64

Amicable Numbers 65

Imperfectly Amicable Numbers 66

Sociable Numbers and Crowds 67

Practical Numbers 68

Baselike Numbers 71

3 Plane Figurate Numbers 73 Polygons 73

Figurate Numbers 74

Triangular Numbers 74

Operations on Triangular Numbers 77

Perfect Numbers, Triangular Numbers and Sums of Cubes 78

Pascal’s Triangle 79

Triangle Inequality Numbers 80

Rectangular Numbers 83

Square Numbers 85

Sums of Square Numbers 88

Positive Square Pair Numbers 89

Bigrade Numbers 90

Pythagorean Triples 90

Primitive Pythagorean Triples 92

Congruent Numbers 93

Fermat’s Last Theorem 94

Happy Numbers 94

Operations on Happy Numbers 95

Happy Number Words 96

Repeating Cycles 97

Patterns in Squares of 1, 11, 111, 98

Squarefree Numbers 99

Tetragonal Numbers 100

Pentagonal Numbers 101

Hexagonal Numbers 103

Recursion and Figurate Numbers 107

Remainder Patterns in Figurate Numbers 109

Gnomic Numbers 109

Lo-Shu Magic Square; Male and Female Numbers 111

4 Solid Figurate Numbers 113 Polyhedra and Solid Figurate Numbers 113

Pyramidal Numbers 115

Tetrahedral Numbers, Triangular Pyramidal Numbers 115

Square Pyramidal Numbers 118

Pentagonal Pyramidal Numbers 120

Hexagonal Pyramidal Numbers 120

Heptagonal and Octagonal Pyramidal Numbers 121

Star Numbers and Star Pyramidal Numbers 121

Rectangular Pyramidal Numbers 122

Cubic Numbers 125

Integers as Sums and Differences of Cubic Numbers, 1729 126

Pythagorean Parallelepiped Numbers 128

5 More Prime Connections 131 Goldbach’s Conjectures 131

Integers as Sums of Odd Integers 133

Integers as Sums of Two Composite Numbers 134

Positive Prime Pair Numbers 134

Prime Line and Prime Circle Numbers 136

Beprisque Numbers 138

A Primes-Between Property 138

Germain Primes 139

Twin Primes 140

Semiprimes and Boolean Integers 140

Snowball Primes 142

Lucky Numbers 142

Prime and Lucky Numbers 144

Polya’s Conjecture about Odd- and Even-Type Integers 145

Balanced Numbers 147

Fermat Numbers 147

Cullen Numbers 148

Ruth–Aaron Numbers 149

6 Digital Patterns and Noteworthy Numbers 151 Monodigit and Repunit Numbers and Langford Sequences 151

Social and Lonely Numbers 153

Additive Multidigital Numbers 155

Multiplicative Multidigital Numbers 156

Kaprekar’s Number 6174, 99 and 1089 157

Doubling Numbers 159

Good Numbers 161

Nearly Good Semiperfect Numbers 163

Powerful Numbers 164

Armstrong Numbers and Digital Invariant Numbers 165

Narcissistic Numbers 167

Additive Digital Root Numbers 167

Additive Persistence of Integers 169

Multiplicative Digital Root Numbers 170

Multiplicative Persistence of Integers 171

Modest and Extremely Modest Numbers 173

Visible Factor Numbers 174

Nude Numbers 175

7 More Patterns and Other Interesting Numbers 177 More Sums of Consecutive Integers 177

Product Patterns for Consecutive Integers 181

Consecutive Integer Divisors 183

Consecutive Number Sums and Square Numbers 184

Factorial Numbers, Applications and Extensions 185

Factorial Sum Numbers and Subfactorial Numbers 188

Hailstone and Ulam Numbers; The Collatz and Ulam Conjecture 189

Palindromic Numbers 191

Creating Palindromic Numbers 193

Palindromic Number Words and Curiosities 194

Palindromic Numbers and Figurate Numbers 196

Palindromic Primes and Emirps 196

Honest Numbers 197

Bell Numbers 198

Catalan Numbers 200

Fibonacci Numbers 203

Lucas Numbers 206

Tribonacci Numbers 207

Tetranacci Numbers 208

Phibonacci Numbers 209

Survivor Numbers or U-Numbers or Ulam Numbers 209

Tautonymic Numbers 210

Lagado Numbers 211

mathematics that is vibrant and motivating Number Treasury3has evolved inorder to serve as a catalyst for those who ascribe to this point of view.

Details

Number Treasury3is a broadening and update of Number Treasury2 The bookcontains information about more than 100 families of positive integers Brief histor-ical notes often accompany the descriptions and examples of the number families

Exercises for each major family are provided to stimulate insight Some exercisescontain problems that are thought provokers to be resolved simply with paper andpencil; others should be tackled with calculator in hand so that lengthier compu-tations can be managed with ease and take the results to a higher level of under-standing Still other problems are intended for more extensive exploration with theuse of computer software In some instances it is helpful to model problems withhands-on materials

The emphasis in Number Treasury3is on doing rather than proving However, thereader is urged to think critically about situations, to provide reasoned explanations,

to make generalizations and to formulate conjectures The book begins with achapter of Investigations These are principally stand-alone activities that representcontent drawn from the Chapters 2 through 7 of the book Their purpose is to set thetone 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 Referencesand recommended readings are supplied so that teachers and students can use thisbook 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 areengaged in learning about number families and who are assigned problems, projects

and papers, Number Treasury3is a useful source of ideas and topics The mix ofdiscussion with examples and illustrations is intended to serve as a writing modelfor the student Both audiences should think critically about the content, providecarefully reasoned explanations, make generalizations, and form conjectures

Who Is Involved

The first edition was completed with the able assistance of six Boston Collegegraduates and undergraduates: Jeanne Cavanaugh, James Cavanaugh, ClaudiaKatze, Stephen Kokoska, Jill Nille, and Jonathan Smith

Seven Boston College graduates and undergraduates were indispensable in theproduction of the second edition Special thanks and grateful appreciation go toJoan Martin for her thoughtful content and style suggestions, editorial advice andword processing skills; to Cynthia Tahlmore, Geraldine Mele, and Erin Mitchellfor computer graphics and word processing assistance; to Allyson Russo, ShannonToomey, and Megan Mazzara for problem solutions

The third edition has been completed by the surviving original author with theinvaluable assistance and perseverance of Geraldine Mele who offered not onlycontent suggestions but who also especially contributed word processing, computergraphics and style expertise Sincere gratitude and appreciation is also extended toJoan Martin for her careful review of the manuscript

What are They?

The Investigations that follow are a set of stand-alone activities Each Investigationfocuses on at least one number family or topic relating to numbers All but six ofthe 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 timerequired 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 thananticipated

Teacher Tips

The Investigations are listed in ascending order of difficulty in the table on the nexttwo pages There are three levels of difficulty represented in the 28 Investigationsthat can be assigned individually or adapted for group work The lowest levelconsists of the first eight Investigations that have the least prerequisites The middlelevel consists of the next nine Investigations and requires more use of abstractreasoning and familiarity with algebraic expressions The final 11 Investigationschallenge the student to persist and probe more deeply in order to complete thework

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 cific Investigations prior to assigning them Assign the Investigations as extendedhomework or as in-class work Some Investigations call for the preparation of

spe-2 Number Treasury3

reports Thus, students may need further directions, especially about the kind ofresources available for them to use Students should know the Internet is an excel-lent 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 Prerequisites & Text reference

4 Footsteps of Lagrange Square numbers, p 85

5,6 Trying Trapezoids Triangular numbers & trapezoids, p 74

7 Hexagons in Black & White Familiarity with recursive & explicit

link Catalan & Pascal, p 200

13 Mysterious Mountains & Binary

Trees

Doing & arranging sketches, p 200

14 Fermat Factorings Prime factorization, p 42

15,16 Factor Lattices LCM, prime factorization, p 46

19,20 Conjecturing with Pascal Articulating patterns, p 79

21 Pythagorean Triple Pursuits Evaluating expressions, p 90

22 Pentagonal Play Squares & triangles within, p 101

23 Triangular Number Turnarounds Visualizing triangles, p 80

24 Centered Triangular Numbers Making algebraic generalizations,

p 7425,26 Catalan Capers Connect algebra & geometry, p 201

27 Highly Composite Numbers Counting divisors, p 45

28,29 Tower of Hanoi & the Reve’s

Puzzle

Recursive actions & thinking, p 56

30 Perfect Number Patterns Using logs to count, p 59

generalizations, p 125

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 3

Page Investigation Prerequisites & Text reference

32 A Medieval Pattern Connect pattern to shape &

number, p 88

33 Prime Magic Trial & error yields results, p 136

34 Dealing with Digits (base ten) Exploring ways to count, p 188

35 Factorial Finishes Importance of 2× 5 in reasoning,

p 185

36 Designing Designs Representation is critical, p 198

37 Fibonacci Fascinations Spreadsheet use, p 203

4 Number Treasury3

Footsteps of Lagrange

Develop the first 20 terms of the sequences n, wheres n is the number of waysn

can be written as a sum of at most 4 squares Note one term will be counted as

How many ways can 9 be written as a sum of at most 4 squares?

2 ways since 9= 22+ 22+ 12and 9= 32

1 Fill in the following table

Ways to Writen as Number

n Sum of at Most 4 Squares of Ways1

234

678

1011121314151617181920

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 inexactly

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 5

Isosceles array Right-angled array

There are many different ways to describe trapezoidal numbers Two of these areshown below

• Subtracting triangular numbers produces trapezoidal numbers

EXAMPLE

Start withT4 SubtractT1andT2.

T4− T1 T4− T2

Thus 7 and 9 are trapezoidal numbers

1 Using subtraction of triangular numbers procedure, name the trapezoidal bers that come from

6 Number Treasury3

2 Name the trapezoidal numbers that consist ofa) 4 rows one of which has 3 dots

b) 5 rows one of which has 4 dots

c) 3 rows one of which hask dots, k > 1.

3 List all ways that each given number can be trapezoidal

4 The trapezoidal number 9 can be pictured in isosceles or right-angled form:

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 thesequence 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.

6 Segments joining the dots in the trapezoidal number 2+ 3 + 4 = 9 form anarray

1 hexagon with 2 dots per side and 2 triangles

Determine that segments joining the arrays of dotsa) 2+ 3 + 4 + 5 + 6 forms 3 hexagons and 6 trianglesb) 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 dotsc) 2+ 3 + · · · + 10

d) 2+ 3 + · · · + 12e) 2+ 3 + · · · + 20

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

Hexagons in Black and White

1 To produce the patterns below you will need a quantity of two different coloredchips Copy each of the designs shown and then create the next two that followthe pattern Use the designs to produce the data in the table Letn 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 chipsC nin rown of the

table That is, expressC nin terms ofC n−1

b) Give an explict formulaf(n) for the total number of chips in terms of n.

3 Create your own patterns using two different colored chips Gather the data andrecord it Summarize your findings with a recursive and an explicit formula

8 Number Treasury3

Marble Art

Polyhedral numbers can be represented physically in a variety of ways Forexample, you can produce effective and attractive models from practice golfballs, holiday glass ball ornaments, or marbles held together with glue Twomodels using marbles are described below Try these and then create somevariations of your own

Square pyramidal numbers

1 Make glued layers of marbles representing the consecutive square numbers Letthem 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 × 3layer, the 2×2 layer, and so on When all layers are glued in place, let the modelstand 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 positionedunder the model to provide interesting reflections

Tetrahedral cluster

1 Build and glue a hexagonal array that has 3 marbles per side Let it stand untilcompletely 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 ontothe indentation of a triangular array of 3 marbles)

Foot

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 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 modelstand until completely dry

Top and bottom view

10 Number Treasury3

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 aspossible 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 followingexample illustrates

If you continue, 4 and four keep repeating.

The number trail has 4 numbers and ends in the honest number 4

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

If the steps are continued, 4 and four keep repeating.

This number trail has five numbers and ends in 4

Find the number trails for:

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

12 Number Treasury3

Geoboard Journeys

Use a geoboard of size 5 pegs by 5 pegs or larger, together with a sheet of squaredot paper for recording results

Your task: Count the number of paths fromA 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 fromA to B.

EXAMPLES

On a 2× 2 geoboard there is 1 path

A B

On a 3× 3 geoboard there are 2 paths

A

B

A

B

In problems 1 and 2,A is the lower left peg and B is the upper right peg.

1 Produce the acceptable paths fromA to B on a

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 youare able the kind of numbers that represent the total path count fromA to B.

3 Using only horizontal or vertical segments and any pegs forB, count all

accept-able paths fromA to B Give the total and describe your counting strategy on a

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 13

Mysterious Mountains and Binary Trees

Mountains are constructed with up-segments and down-segments

1 mountain design ispossible using one up–

one down-segment

2 mountain designs are possible using twoup–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

14 Number Treasury3

Fermat Factorings

Pierre de Fermat (1601–1665) a French lawyer by profession and an amateurmathematician 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 toN until a square was

reached If the given number is even, first divide by 2 until an odd non-squareintegerN appears.

EXAMPLE

Find the prime factorization of 105

105 Since 121 is a square, then 105 can be written as+ 1 (11 +−)(11 −−), where−is replaced by the

106 number of odd integers needed to reach a perfect

+ 3 square In this case, 4 odd integers are added

Find the prime factorization of 84

Since 84 is even, divide by 2 until an odd non-square integer appears

2 Explain why Fermat’s algorithm works

3 Write a report on the life of Pierre de Fermat, including names of his ical friends and some of the mathematical problems he worked on Be sure toinclude information about Fermat primes, Fermat’s Last Theorem, and AndrewWiles

mathemat-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 linesegments, in linear, two, three, and higher dimensional arrays Such figures arecalled factor lattices and each vertex of these figures is labeled with one of thedivisors of the integer Factor lattices can be produced using prime factorizationand 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= 34is:

0

The factor lattice for 81 has 5 vertices, one for each divisor of 81 Also, the factorlattice 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 27has eight vertices and seven unit segments

1 The factor lattices for 75 and 36 are shown below Note that lcm is an ation for least common multiple Study them carefully for clues.

abbrevi-lcm(3,2)

lcm(6,4)

lcm(18,12) lcm(9,6)

16 Number Treasury3

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 verticesand edges of a cube

2 6 lcm(2,3)

5 15 lcm(3,5)

30 lcm(2,3,5)

31

10lcm(2,5)

Try building models of three dimensional factor lattices You can use toothpicks as

edges with gum drops or mini-marshmallows for vertices or commercially availablematerials

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 thenumber 210

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 17

A Juggling Act

You have two unmarked containers that you take to a well

One container holdsa 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 accomplishthis 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 ofwater? Explain your answer

6 Whena 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 haveinto two equal amounts

18 Number Treasury3

The Super Sum: 1 + 2 + 3 + · · · · · · · · · + nnn

As a young boy, Carl Friedrich Gauss (1777–1855) attended St Katherine’sVolksschule, a one-room school in Brunswick, Germany The story is told thatwhen he was about eight years old, his teacher asked the class to compute the sum

1+ 2 + 3 + · · · + 100 Carl was able to get the solution quickly and was the first

to place his slate with the correct answer on the table

1 Solve Carl’s problem Describe in detail how you arrived at your total

2 Use your reasoning in problem 1 to computea) 1+ 2 + 3 + · · · + 99 b) 1+ 2 + 3 + · · · + 101c) 1+ 2 + 3 + · · · + 500 d) 1+ 2 + 3 + · · · + 175

3 Generalize your reasoning in problem 2 to find the total of the super sum

6 Find the rectangle count for the grids of size

7 Challenge Use the super sum total to find the total rectangle count for grids of

size

a) 5× n b)m × n

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 19

Conjecturing with Pascal

Diagonal 1

Diagonal 2

Diagonal 3

The Pascal Triangle

A conjecture is a statement that appears to be true based on the evidence at hand In

the Pascal triangle investigations below, you will get an opportunity to formulateyour own conjectures

1 Calculate the row sums for the rows 0 through 8 Look for patterns to completethe conjecture: In rown, the row sum is Calculate the cumulative row sums

for rows 0 through 8 Complete the conjecture: The cumulative row sum forrows 0 throughn is

2 a) Alternate+/− between the numbers in each row 0 through 8 Then calculatethe sums For example, 1+4 −6 + 4 − 1 = 2 Look for patterns to completethe conjecture: In rown, the sum is

b) Alternate –/+ between the numbers in each row 0 through 8 Then calculatethe sums For example, 1− 4 +6 −4 +1 = 0 Look for patterns to completethe conjecture: In rown, the sum is

3 a) In row 0 the single number is odd; in rows 1 and 3 each number is odd Namethe next 5 rows in which each number is odd

b) In rows 2 and 4, except for the two 1s, the remaining numbers are even

Name the next 5 rows in which each number, except for the two 1s, is even

Explain why this happens

c) Based on your findings make a conjecture that indicates which rows containall odd numbers and which rows except for the two 1s contain all evennumbers

20 Number Treasury3

4 Study the hexagonal grid on which the first few rows of the Pascal triangleappear Without first filling in the numbers in more rows of the triangle, shadethe hexagons in each row as follows If a hexagon represents an odd numbershade it, and if a hexagon represents an even number do not shade it Use thesefacts:

a) The sum of two even numbers is an even number

b) The sum of two odd numbers is an even number

c) The sum of an even and an odd number is an odd number

A triangular design should result By counting unshaded hexagons, identify thevarious triangular numbers that appear in your design

5 Find references that have information on the Sierpinski triangle Prepare a reportthat describes how this triangle is formed and how it is related to your Pascaltriangle patterns Find out about the Chaos Game How does it work? How is itconnected to these designs?

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 21

Pythagorean Triple Pursuits

A Pythagorean triple is a collection of three positive integersa, b, c such that

a2+ b2= c2

1 a) Use a spreadsheet to list 50 different values fora, b, and c where the n’s are

positive integers and

a = 2n + 1, b = 2n2+ 2n, c = 2n2+ 2n + 1.

b) Show thata2+ b2= c2for the values ofa, b, and c in 1(a).

2 a) Use a spreadsheet to produce three lists of 50 different values fora, b, and c

wherem, n are positive integers and m > n and

a = 2mn, b = m2− n2, c = m2+ n2.Start withn = 1 and let m = n + 1; m = n + 2; and m = n + 3 Choose other

combinations form, n and use the spreadsheet to list other sets of 50 different

values

b) Show thata2+ b2= c2for the values ofa, b, and c in 2(a).

3 Consult numerous paper and Online resources that treat mathematics content

as well as the history of mathematics

a) Write a brief report about Pythagorean triples that includes an explanation

of the difference between a Pythagorean triple and a primitive Pythagorean

triple

b) In your report include several formulas that produce primitive Pythagorean

triples Find out when and where the formulas originated

4 There are 50 Pythagorean triples in which each number is less than 100 Make

a list of the ones you find Discuss the patterns you observe in and among thetriples

22 Number Treasury3

Pentagonal Play

1 The first five pentagonal numbers are:

P1= 1, P2 = 5, P3 = 12, P4 = 22, P5 = 35.

a) Name the next five pentagonal numbersP6throughP10.

b) Find a formula that gives thenth pentagonal number P n

2 The pentagonal numbers greater than 1 can be represented in a “houselike form”

as the sum of dot arrays that are a combination of a square plus a triangle array

Sketch the “houselike forms” forP5,P6,P7 ExpressP8,P9,P10as the sum of

a square and triangular number ExpressP nas the sum of a particular squareand triangular number

3 The triangular numbers areT1= 1, T2 = 3, T3 = 6, with T n = n(n + 1)/2.

Observe thatP2 = 5 = 3 + 1 + 1 and P3 = 12 = 6 + 3 + 3 Write each ofthe pentagonal numbersP4throughP10as the sum of three triangular numbers

Show that in generalP ncan be written as the sum of three triangular numbers

4 Compute the sum of the digits of each pentagonal number through P20 Ifnecessary, keep adding to reduce the sum to a single digit This digit is calledthe digital sum Which digits occur as digital sums? Make a conjecture aboutthe patterns you observe

5 List the units or ones digit of each of the first 25 pentagonal numbers Whichdigits appear in the list? Make a conjecture about the patterns you observe

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 23

Triangular Number Turnarounds

A triangular number is a number that is the sum of consecutive numbers starting

3 Triangular numbers can be represented by triangular arrays of dots or chips

Determine the minimum number of dot or chip moves required to turn the triangle from a point up to a point down position.

Develop a strategy and draw the sketches to turn aroundT4, T5, , T10.

4 Make a spreadsheet or calculator listing of the first 50 triangular numbers

Use this data to estimate if there are just as many even triangular numbers asodd triangular numbers amongst the first 100 triangular numbers Extend yourspreadsheet to verify your guess

24 Number Treasury3

Centered Triangular Numbers

The number below each figure is its dot count

1 Draw the next figure in the sequence of centered triangular numbers, the work

of Jordanus de Nemore from Germany in the 13th century, and give its dotcount

2 Complete the table below by naming the next four elements of the sequences n

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 25

Catalan Capers

Eugene Charles Catalan (1814–1894), a Belgian mathematician, had a specialsequence named for him even though he did not discover the sequence He did,however, establish results about it It is Leonhard Euler (1707–1783) who is creditedwith finding the sequence about 100 years before Catalan’s work

Call the Catalan numbersC nforn = 1, 2, 3, and so on C1= 1 and C2= 2 Namethe Catalan numbersC3andC4by solving this problem

In how many ways can a convex polygon be dissected into nonoverlapping triangles

by drawing diagonals that do not intersect?

LetC n represent the number of ways a convex polygon withn + 2 sides can be

dissected into nonoverlapping triangles SetC1= 1

1 Find C3 by sketching the number of different ways a convex pentagon can

be dissected into nonoverlapping triangles Then, carefully study the examplesdone for you, and in each case label the diagonals and the bottom side ofthe pentagon with a matching expression (Hint: you will not need to use allthe pentagons All expressions should reada b c d with parentheses inserted

appropriately.)

26 Number Treasury3

2 FindC4by sketching the number of different ways a convex hexagon can bedissected into nonoverlapping triangles Study the example done for you and ineach case label the diagonals and bottom side of the hexagon with a matchingexpression (Hint: you will not need to use all the hexagons All expressionsshould reada b c d e with parentheses inserted appropriately.)

b

c d

e

cd

(c d )e ab

(ab)((cd)e) a

b

c d

e a

b

c d

e a

b

c d

e

a b

c d

e a b

c d

e a b

c d e a

b

c d e

a

b

c d

e a

b

c d

e a

b

c d

e a

b

c d

e

a

b

c d

b

c d

b

c d

e a

b

c d

e

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 27

Highly Composite Numbers

A highly composite positive integern > 1 is a positive integer that has more

divisors than any positive integer less thann.

Srinivasa Ramanujan (1887–1920), an outstanding number theorist from India,worked with all kinds of number patterns When the accomplished British mathe-matician Godfrey Hardy (1877–1947) rated himself and some of his colleagues on

a scale of math ability from 0 to 100, he gave himself a score of 25 and Ramanujan

a 100 Highly composite numbers were one of Ramanujan’s inventions

The number 2 is highly composite because it is the first number to have two divisors,

namely 1, 2 Two is the only prime highly composite number

The number 3 is not highly composite because it does not have more than twodivisors

The number 4 is highly composite because it is the first number to have three

k thenN has (a1+ 1)(a2 + 1) (a k + 1) divisors.

Thep is are primes, and thea is are the corresponding powers

For example, consider 30 and 45

28 Number Treasury3

The Tower of Hanoi and the Reve’s Puzzle

In 1883 Edouard Lucas, a French mathematician who worked with Fibonacci-likesequences and perfect numbers, introduced (under the pseudonym N Claus) the

puzzle he called the Tower of Hanoi Today this puzzle is known under a variety

of names and stories to many who are interested in mathematics and puzzles

However, the basic idea of the puzzle is constant You have a pyramid ofn disks as

shown below You must transfer the pyramid from one of three spikes to anotherusing the least number of moves

The conditions are as follows:

a) you transfer 1 disk per move

b) you may not place a larger disk on top of a smaller disk

1 Complete the table:

3 If it takes 1 second to move 1 disk, how long does it take to move a pyramid of

a) 10 disks? Give your result in minutes

b) 15 disks? Give your result in hours

c) 20 disks? Give your result in days

d) 30 disks? Give your result in years (let 1 year be 365 days)

e) 64 disks? Give your result in years

A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 29

In 1907 the well-known English puzzlist Henry Dudeney published The

bury Puzzles This collection of problems based on Geoffrey Chaucer’s bury Tales contained the Reve’s Puzzle.

Canter-In Dudeney’s narrative the reve asked the traveler: If a stack of eight cheesesgraduated in size and arranged top to bottom from smallest to largest were placed

on one of four stools, what is the least number of moves needed to transfer thestack of cheeses to another stool?

The rules require that:

a) you transfer 1 cheese per move

b) you may not place a larger cheese on top of a smaller cheese

Thus, this puzzle is an extension of the Tower of Hanoi puzzle from three spikes

to four spikes

Use the rules described above to determine the following solutions

4 Complete the table:

Least No of Moves

5 Can you describe your answer for 4 cheeses recursively in terms of your answerfor 3 cheeses? How about your answer for 5 cheeses recursively in terms ofyour answer for 4 cheeses?

6 Dudeney asked for solutions for 8, 10, and 21 cheeses Using the table can youdetermine solutions for the cases of 10 and 21 cheeses?

30 Number Treasury3

Perfect Number Patterns

A perfect number is a positive integer that equals the sum of its proper divisors.

A proper divisor of a number is any divisor of the number excepting itself Thefirst perfect number is 6 6= 1 + 2 + 3

1 All perfect numbers found to date are even and can be expressed by a formulagiven by Euclid: 2n−1 (2 n − 1), where 2 n− 1 is prime The formula does notproduce a perfect number for alln Determine which values of n = 2, 3, 4, 5, 6,

7, 8, 9, 10, 11, 12 give perfect numbers List the corresponding perfect numbers

The number of digits in a number is 1 more than the characteristic in the commonlog (log10) expression of the number.

Recall, loga b = b log a Use log 2 = 0.30103.

Since the characteristic is 2, the number of digits is 2+ 1 = 3

2 Use a calculator to determine the number of digits in the following perfectnumbers:

Write 496 and 8128 as sums of powers of 2

4 Use the fact that the geometric series

1+ 2 + 22+ · · · + 2n−1= 2n− 1 to show that all perfect numbers of the form

2n−1 (2 n − 1) are sums of powers of 2.

5 Use a number theory resource or the Internet to learn more about perfect bers Write a brief paper that describes and explains other facts about perfectnumbers

num-A Perfect Number of Investigations 28= 1 + 2 + 4 + 7 + 14 31

1 Build the next three models that follow in the sequence

2 Use the models to gather data Complete the table

Perimeter of Front Volume in Total Surface AreaModel Face in Units Cubic Units in Square Units

a) Give a recursive relation for thenth term that uses the preceding term or

terms of the sequence

b) Give an explicit relation for thenth term in terms of n.

32 Number Treasury3

A Medieval Pattern

The number of unit squares in each figure is listed under the figure

1 Use grid paper and shade in the next figure in the sequence and give the number

of unit squares,s5, in the figure.

2 Complete the table below by listing the next five terms of the sequences n

3 The terms,s n, in the table are odd numbers Explain in a sentence or two whythe pattern produces only odd numbers of unit squares

4 Find a recursive formula for the unit square count,s n

5 Find an explicit formula for the unit square count,s n

6 Create your own medieval pattern using unit squares Lets nbe the unit squarecount

a) Draw the first four figures on grid paper

b) Prepare a table that displays the first ten terms of your sequences n.c) Give a recursive or explicit formula fors n