**MSRI Mathematical Circles Library**

Volume: 23;
2019;
262 pp;
Softcover

MSC: Primary 00;

**Print ISBN: 978-1-4704-5115-8
Product Code: MCL/23**

List Price: $25.00

AMS Member Price: $20.00

MAA Member Price: $22.50

**Electronic ISBN: 978-1-4704-5328-2
Product Code: MCL/23.E**

List Price: $25.00

AMS Member Price: $20.00

MAA Member Price: $22.50

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#### Supplemental Materials

# How Round Is a Cube?: And Other Curious Mathematical Ponderings

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*James Tanton*

A co-publication of the AMS and the Mathematical Sciences Research Institute

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.

#### Reviews & Endorsements

'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

#### Table of Contents

# Table of Contents

## How Round Is a Cube?: And Other Curious Mathematical Ponderings

- Cover Cover11
- Title page iii4
- Preface xi12
- Topics Explored xiii14
- Essay 1. Dragons and Poison 116
- Essay 2. Folding Tetrahedra 520
- Essay 3. The Arbelos 924
- Essay 4. Averages via Distances 1732
- Essay 5. Ramsey Theory 2338
- Essay 6. Inner Triangles 2944
- Essay 7. Land or Water? 3752
- Essay 8. Escape 4560
- Essay 9. Flipping a Coin for a Year 5368
- Essay 10. Coinciding Digits 5974
- Essay 11. Inequalities 6378
- Essay 12. Gauss’s Shoelace Formula 6782
- 12.1. Step 1: Nicely Situated Triangles 7186
- 12.2. Step 2: General Triangles 7388
- 12.3. Step 3: Begin Clear of the Effect of Motion 7489
- 12.4. Step 4: Being Clear on Starting Points 7590
- 12.5. Step 5: Steps 1 and 2 Were Unnecessary! 7691
- 12.6. Step 6: Quadrilaterals 7691
- 12.7. Step 7: Beyond Quadrilaterals 7893

- Essay 13. Subdividing a Square into Triangles 8196
- Essay 14. Equilateral Lattice Polygons 89104
- Essay 15. Broken Sticks and Viviani’s Theorem 99114
- Essay 16. Viviani’s Converse? 109124
- Essay 17. Integer Right Triangles 117132
- Essay 18. One More Question about Integer Right Triangles 127142
- Essay 19. Intersecting Circles 131146
- Essay 20. Counting Triangular and Square Numbers 141156
- Essay 21. Balanced Sums 149164
- Essay 22. The Prouhet–Thue–Morse Sequence 159174
- Essay 23. Some Partition Numbers 169184
- Essay 24. Ordering Colored Fractions 177192
- Essay 25. How Round Is a Cube? 187202
- Essay 26. Base and Exponent Switch 201216
- Essay 27. Associativity and Commutativity Puzzlers 207222
- Essay 28. Very Triangular and Very Very Triangular Numbers 215230
- Essay 29. Torus Circles 219234
- Essay 30. Trapezoidal Numbers 231246
- Essay 31. Square Permutations 239254
- Essay 32. Tupper’s Formula 245260
- Essay 33. Compositional Square Roots 251266
- Essay 34. Polynomial Permutations 257272
- Back Cover Back Cover1280