Softcover ISBN:  9781470451158 
Product Code:  MCL/23 
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AMS Member Price:  $28.00 
eBook ISBN:  9781470453282 
Product Code:  MCL/23.E 
List Price:  $30.00 
MAA Member Price:  $27.00 
AMS Member Price:  $24.00 
Softcover ISBN:  9781470451158 
eBook: ISBN:  9781470453282 
Product Code:  MCL/23.B 
List Price:  $65.00$50.00 
MAA Member Price:  $58.50$45.00 
AMS Member Price:  $52.00$40.00 
Softcover ISBN:  9781470451158 
Product Code:  MCL/23 
List Price:  $35.00 
MAA Member Price:  $31.50 
AMS Member Price:  $28.00 
eBook ISBN:  9781470453282 
Product Code:  MCL/23.E 
List Price:  $30.00 
MAA Member Price:  $27.00 
AMS Member Price:  $24.00 
Softcover ISBN:  9781470451158 
eBook ISBN:  9781470453282 
Product Code:  MCL/23.B 
List Price:  $65.00$50.00 
MAA Member Price:  $58.50$45.00 
AMS Member Price:  $52.00$40.00 

Book DetailsMSRI Mathematical Circles LibraryVolume: 23; 2019; 262 ppMSC: Primary 00;
This book is a collection of 34 curiosities, each a quirky and delightful gem of mathematics and each a shining example of the joy and surprise that mathematics can bring. Intended for the general math enthusiast, each essay begins with an intriguing puzzle, which either springboards into or unravels to become a wondrous piece of thinking. The essays are selfcontained and rely only on tools from highschool mathematics (with only a few pieces that eversobriefly brush up against highschool calculus).
The gist of each essay is easy to pick up with a cursory glance—the reader should feel free to simply skim through some essays and dive deep into others. This book is an invitation to play with mathematics and to explore its wonders. Much joy awaits!
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.ReadershipMath circles students and organizers, participants and organizers of math summer camps for highschool students, and anyone interested in learning or teaching mathematics at the highschool level.

Table of Contents

Cover

Title page

Preface

Topics Explored

Essay 1. Dragons and Poison

1.1. Analyzing the Puzzle

Essay 2. Folding Tetrahedra

2.1. Polyhedron Symmetry

Essay 3. The Arbelos

3.1. The Arbelos

3.2. The Area of the Arbelos

3.3. The Archimedean Circles

3.4. The Other Common Tangent Segment

3.5. Conic Curves

Essay 4. Averages via Distances

4.1. Ideal vs. Real Data

4.2. Using the Distance Formula: Euclidean Distance

4.3. Using the Taxicab Metric

4.4. Mean, Median, and …

Essay 5. Ramsey Theory

5.1. Ramsey Theory

5.2. Connections to the Opening Puzzler

Essay 6. Inner Triangles

6.1. Two Observations about Triangles

6.2. Routh’s Theorem

6.3. Solving Feynman’s Problem without the Big Guns

Essay 7. Land or Water?

7.1. The Answer to the First Puzzler

7.2. Towards Answering the Second Puzzler

7.3. Proving the Theorem

Essay 8. Escape

8.1. Leibniz’s Harmonic Triangle

8.2. The Infinite Stocking Property

8.3. Variations of Leibniz’s Harmonic Triangle

8.4. Solving the Opening Puzzle

Essay 9. Flipping a Coin for a Year

9.1. Solutions to Coin Tossing 1

9.2. Solutions to Coin Tossing 2

Essay 10. Coinciding Digits

10.1. The Chinese Remainder Theorem

10.2. The Opening Puzzler

Essay 11. Inequalities

11.1. Solving Puzzle 1

Essay 12. Gauss’s Shoelace Formula

12.1. Step 1: Nicely Situated Triangles

12.2. Step 2: General Triangles

12.3. Step 3: Begin Clear of the Effect of Motion

12.4. Step 4: Being Clear on Starting Points

12.5. Step 5: Steps 1 and 2 Were Unnecessary!

12.6. Step 6: Quadrilaterals

12.7. Step 7: Beyond Quadrilaterals

Essay 13. Subdividing a Square into Triangles

13.1. Sperner’s Lemma

13.2. The Impossibility Proof

13.3. Case 1: 𝑁 is an Odd Integer

13.4. Case 2: 𝑁 is an Even Integer

Essay 14. Equilateral Lattice Polygons

14.1. Areas of Lattice Polygons

14.2. Aside: Pick’s Theorem

14.3. Equilateral Lattice Polygons

14.4. The Answer with Cheating

14.5. Dots of Zero Width

Essay 15. Broken Sticks and Viviani’s Theorem

15.1. Viviani’s Theorem

15.2. Broken Sticks and Triangles

15.3. Something Unsettling

15.4. Focus on the Left Piece

15.5. Summing Probabilities

15.6. Answers

Essay 16. Viviani’s Converse?

16.1. Planes above Triangles

16.2. The Equation of a Plane

16.3. The Distance of a Point from a Line

16.4. The Converse of Viviani’s Theorem

16.5. Other Figures

Essay 17. Integer Right Triangles

17.1. A Cute Way to Find Pythagorean Triples

17.2. A Primitive Tidbit

17.3. The Answers to All the Curiosities

Essay 18. One More Question about Integer Right Triangles

18.1. A Precursor Question

18.2. The Answer to the Main Question

Essay 19. Intersecting Circles

19.1. Loops on a Page

19.2. Circles on a Page

19.3. Polygons on a Page

Essay 20. Counting Triangular and Square Numbers

20.1. Some Interplay between Square and Triangular Numbers

20.2. Formulas

20.3. Counting Figurate Numbers

20.4. The Squangular Numbers

20.5. Something Bizarre!

Essay 21. Balanced Sums

21.1. On Sums of Consecutive Counting Numbers

21.2. How to Find More!

21.3. To Summarize

Essay 22. The Prouhet–Thue–Morse Sequence

22.1. The Prouhet–Thue–Morse Sequence

22.2. The Opening Puzzler

22.3. Alternative Constructions

22.4. Proving the Puzzler

Essay 23. Some Partition Numbers

23.1. The Partition Numbers

23.2. Partitions into a Fixed Number of Parts

23.3. Cracking the 𝑃₃(𝑛) Formula

Essay 24. Ordering Colored Fractions

24.1. Coloring and Ordering Fractions

24.2. A Side Track

24.3. The Mediant

24.4. Explaining Colored Fractions

24.5. A Bad Example?

24.6. Addendum

Essay 25. How Round Is a Cube?

25.1. Deficiencies in Surface Circles

25.2. Rounding the Cube

25.3. A Better Way to Count Total Pointiness

25.4. Shaving Corners Does Not Help!

25.5. Not All Shapes Are Spherelike!

25.6. Christopher Columbus and Others

Essay 26. Base and Exponent Switch

26.1. The Graph of 𝑦=𝑥^{1/𝑥} for 𝑥>0

26.2. The Graph of 𝑥^{𝑦}=𝑦^{𝑥} for 𝑥>0, 𝑦>0

26.3. A Connection to 𝑤^{𝑤^{𝑤^{𝑤^{\adots}}}}

26.4. Appendix: A Tricky Swift Proof

Essay 27. Associativity and Commutativity Puzzlers

27.1. Order

27.2. Explaining the Puzzler and Its Variations

Essay 28. Very Triangular and Very Very Triangular Numbers

28.1. Numbers in Binary

28.2. Very Triangular Numbers

Essay 29. Torus Circles

29.1. The Equation of a Torus

29.2. Slicing an Ideal Bagel: The Puzzler

29.3. The Four Circles Property

Essay 30. Trapezoidal Numbers

30.1. Trapezoidal Numbers

30.2. From Rectangles to Trapezoids

30.3. Two Transformations

30.4. Counting Presentations

Essay 31. Square Permutations

31.1. Square Permutations

31.2. Taking Stock

Essay 32. Tupper’s Formula

32.1. Understanding the Notation

32.2. The mod Function in Computer Science

32.3. Encoding a Picture

32.4. Delightful Recursive Quirkiness

32.5. How to Find a Particular Picture

32.6. Tupper’s Dimensions

Essay 33. Compositional Square Roots

33.1. Compositional Square Roots

33.2. Constructing Compositional Square Roots

Essay 34. Polynomial Permutations

34.1. Playing with Polynomials

Back Cover


Additional Material

Reviews

'How Round is a Cube?' is a charming collection of mathematical 'ponderings' neatly organized into 34 essays. The text was published as part of the MSRI Mathematical Circles Library series, and the text certainly sparks a creative, playful approach to mathematics that is appropriate for a math circle (or math teachers circle). However, the author's motivation is not purely to prompt such circles  and he himself admits that the collection and cataloging of these problems stems primarily from an interest in (and a joy of) mathematical gems with quirky twists. The result is a thoughtprovoking book that will pique the interest of math circle participants and solo math enthusiasts.
John Ross, Southwestern University


RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
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This book is a collection of 34 curiosities, each a quirky and delightful gem of mathematics and each a shining example of the joy and surprise that mathematics can bring. Intended for the general math enthusiast, each essay begins with an intriguing puzzle, which either springboards into or unravels to become a wondrous piece of thinking. The essays are selfcontained and rely only on tools from highschool mathematics (with only a few pieces that eversobriefly brush up against highschool calculus).
The gist of each essay is easy to pick up with a cursory glance—the reader should feel free to simply skim through some essays and dive deep into others. This book is an invitation to play with mathematics and to explore its wonders. Much joy awaits!
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.
Math circles students and organizers, participants and organizers of math summer camps for highschool students, and anyone interested in learning or teaching mathematics at the highschool level.

Cover

Title page

Preface

Topics Explored

Essay 1. Dragons and Poison

1.1. Analyzing the Puzzle

Essay 2. Folding Tetrahedra

2.1. Polyhedron Symmetry

Essay 3. The Arbelos

3.1. The Arbelos

3.2. The Area of the Arbelos

3.3. The Archimedean Circles

3.4. The Other Common Tangent Segment

3.5. Conic Curves

Essay 4. Averages via Distances

4.1. Ideal vs. Real Data

4.2. Using the Distance Formula: Euclidean Distance

4.3. Using the Taxicab Metric

4.4. Mean, Median, and …

Essay 5. Ramsey Theory

5.1. Ramsey Theory

5.2. Connections to the Opening Puzzler

Essay 6. Inner Triangles

6.1. Two Observations about Triangles

6.2. Routh’s Theorem

6.3. Solving Feynman’s Problem without the Big Guns

Essay 7. Land or Water?

7.1. The Answer to the First Puzzler

7.2. Towards Answering the Second Puzzler

7.3. Proving the Theorem

Essay 8. Escape

8.1. Leibniz’s Harmonic Triangle

8.2. The Infinite Stocking Property

8.3. Variations of Leibniz’s Harmonic Triangle

8.4. Solving the Opening Puzzle

Essay 9. Flipping a Coin for a Year

9.1. Solutions to Coin Tossing 1

9.2. Solutions to Coin Tossing 2

Essay 10. Coinciding Digits

10.1. The Chinese Remainder Theorem

10.2. The Opening Puzzler

Essay 11. Inequalities

11.1. Solving Puzzle 1

Essay 12. Gauss’s Shoelace Formula

12.1. Step 1: Nicely Situated Triangles

12.2. Step 2: General Triangles

12.3. Step 3: Begin Clear of the Effect of Motion

12.4. Step 4: Being Clear on Starting Points

12.5. Step 5: Steps 1 and 2 Were Unnecessary!

12.6. Step 6: Quadrilaterals

12.7. Step 7: Beyond Quadrilaterals

Essay 13. Subdividing a Square into Triangles

13.1. Sperner’s Lemma

13.2. The Impossibility Proof

13.3. Case 1: 𝑁 is an Odd Integer

13.4. Case 2: 𝑁 is an Even Integer

Essay 14. Equilateral Lattice Polygons

14.1. Areas of Lattice Polygons

14.2. Aside: Pick’s Theorem

14.3. Equilateral Lattice Polygons

14.4. The Answer with Cheating

14.5. Dots of Zero Width

Essay 15. Broken Sticks and Viviani’s Theorem

15.1. Viviani’s Theorem

15.2. Broken Sticks and Triangles

15.3. Something Unsettling

15.4. Focus on the Left Piece

15.5. Summing Probabilities

15.6. Answers

Essay 16. Viviani’s Converse?

16.1. Planes above Triangles

16.2. The Equation of a Plane

16.3. The Distance of a Point from a Line

16.4. The Converse of Viviani’s Theorem

16.5. Other Figures

Essay 17. Integer Right Triangles

17.1. A Cute Way to Find Pythagorean Triples

17.2. A Primitive Tidbit

17.3. The Answers to All the Curiosities

Essay 18. One More Question about Integer Right Triangles

18.1. A Precursor Question

18.2. The Answer to the Main Question

Essay 19. Intersecting Circles

19.1. Loops on a Page

19.2. Circles on a Page

19.3. Polygons on a Page

Essay 20. Counting Triangular and Square Numbers

20.1. Some Interplay between Square and Triangular Numbers

20.2. Formulas

20.3. Counting Figurate Numbers

20.4. The Squangular Numbers

20.5. Something Bizarre!

Essay 21. Balanced Sums

21.1. On Sums of Consecutive Counting Numbers

21.2. How to Find More!

21.3. To Summarize

Essay 22. The Prouhet–Thue–Morse Sequence

22.1. The Prouhet–Thue–Morse Sequence

22.2. The Opening Puzzler

22.3. Alternative Constructions

22.4. Proving the Puzzler

Essay 23. Some Partition Numbers

23.1. The Partition Numbers

23.2. Partitions into a Fixed Number of Parts

23.3. Cracking the 𝑃₃(𝑛) Formula

Essay 24. Ordering Colored Fractions

24.1. Coloring and Ordering Fractions

24.2. A Side Track

24.3. The Mediant

24.4. Explaining Colored Fractions

24.5. A Bad Example?

24.6. Addendum

Essay 25. How Round Is a Cube?

25.1. Deficiencies in Surface Circles

25.2. Rounding the Cube

25.3. A Better Way to Count Total Pointiness

25.4. Shaving Corners Does Not Help!

25.5. Not All Shapes Are Spherelike!

25.6. Christopher Columbus and Others

Essay 26. Base and Exponent Switch

26.1. The Graph of 𝑦=𝑥^{1/𝑥} for 𝑥>0

26.2. The Graph of 𝑥^{𝑦}=𝑦^{𝑥} for 𝑥>0, 𝑦>0

26.3. A Connection to 𝑤^{𝑤^{𝑤^{𝑤^{\adots}}}}

26.4. Appendix: A Tricky Swift Proof

Essay 27. Associativity and Commutativity Puzzlers

27.1. Order

27.2. Explaining the Puzzler and Its Variations

Essay 28. Very Triangular and Very Very Triangular Numbers

28.1. Numbers in Binary

28.2. Very Triangular Numbers

Essay 29. Torus Circles

29.1. The Equation of a Torus

29.2. Slicing an Ideal Bagel: The Puzzler

29.3. The Four Circles Property

Essay 30. Trapezoidal Numbers

30.1. Trapezoidal Numbers

30.2. From Rectangles to Trapezoids

30.3. Two Transformations

30.4. Counting Presentations

Essay 31. Square Permutations

31.1. Square Permutations

31.2. Taking Stock

Essay 32. Tupper’s Formula

32.1. Understanding the Notation

32.2. The mod Function in Computer Science

32.3. Encoding a Picture

32.4. Delightful Recursive Quirkiness

32.5. How to Find a Particular Picture

32.6. Tupper’s Dimensions

Essay 33. Compositional Square Roots

33.1. Compositional Square Roots

33.2. Constructing Compositional Square Roots

Essay 34. Polynomial Permutations

34.1. Playing with Polynomials

Back Cover

'How Round is a Cube?' is a charming collection of mathematical 'ponderings' neatly organized into 34 essays. The text was published as part of the MSRI Mathematical Circles Library series, and the text certainly sparks a creative, playful approach to mathematics that is appropriate for a math circle (or math teachers circle). However, the author's motivation is not purely to prompt such circles  and he himself admits that the collection and cataloging of these problems stems primarily from an interest in (and a joy of) mathematical gems with quirky twists. The result is a thoughtprovoking book that will pique the interest of math circle participants and solo math enthusiasts.
John Ross, Southwestern University