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How Round Is a Cube? : And Other Curious Mathematical Ponderings
 
James Tanton Mathematical Association of America, Washington, DC
A co-publication of the AMS and Mathematical Sciences Research Institute
How Round Is a Cube?
Softcover ISBN:  978-1-4704-5115-8
Product Code:  MCL/23
List Price: $35.00
MAA Member Price: $31.50
AMS Member Price: $28.00
eBook ISBN:  978-1-4704-5328-2
Product Code:  MCL/23.E
List Price: $30.00
MAA Member Price: $27.00
AMS Member Price: $24.00
Softcover ISBN:  978-1-4704-5115-8
eBook: ISBN:  978-1-4704-5328-2
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
How Round Is a Cube?
Click above image for expanded view
How Round Is a Cube? : And Other Curious Mathematical Ponderings
James Tanton Mathematical Association of America, Washington, DC
A co-publication of the AMS and Mathematical Sciences Research Institute
Softcover ISBN:  978-1-4704-5115-8
Product Code:  MCL/23
List Price: $35.00
MAA Member Price: $31.50
AMS Member Price: $28.00
eBook ISBN:  978-1-4704-5328-2
Product Code:  MCL/23.E
List Price: $30.00
MAA Member Price: $27.00
AMS Member Price: $24.00
Softcover ISBN:  978-1-4704-5115-8
eBook ISBN:  978-1-4704-5328-2
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 Details
     
     
    MSRI Mathematical Circles Library
    Volume: 232019; 262 pp
    MSC: 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 self-contained and rely only on tools from high-school mathematics (with only a few pieces that ever-so-briefly brush up against high-school 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.

    Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

    Readership

    Math circles students and organizers, participants and organizers of math summer camps for high-school students, and anyone interested in learning or teaching mathematics at the high-school 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 Sphere-like!
    • 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
  • 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 thought-provoking book that will pique the interest of math circle participants and solo math enthusiasts.

      John Ross, Southwestern University
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 232019; 262 pp
MSC: 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 self-contained and rely only on tools from high-school mathematics (with only a few pieces that ever-so-briefly brush up against high-school 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.

Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

Readership

Math circles students and organizers, participants and organizers of math summer camps for high-school students, and anyone interested in learning or teaching mathematics at the high-school 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 Sphere-like!
  • 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 thought-provoking book that will pique the interest of math circle participants and solo math enthusiasts.

    John Ross, Southwestern University
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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