Instead of treating mathematics as a collection of separate topics or isolated courses, this book highlights the patterns and relationships that appear throughout mathematics. Rather than keeping studies confined to a single field, the puzzles here look beyond boundaries and discover links among different areas of knowledge. Within the domain of mathematics, this book focuses on the fun of finding such connections through puzzles.
Math Affinity Puzzles uses analogies to explore connections between mathematical terms and concepts, where solving a puzzle requires unraveling the relationships between the mathematical ideas. Solving this kind of mathematical puzzle is thus not about assessing understanding but deepening it, making the connections, and having fun doing it.
Because there may be multiple relationships present, one answer is not claimed to be correct. Instead, which answer is best comes from one point of view. The short descriptions follow, touching on the different choices while going more deeply into historical or other aspects of the relationships.
The puzzles will deepen the reader’s understanding and appreciation of, as well as foster discussion about, the relationships among mathematical ideas.
The puzzles in this book can be enjoyed on their own or as supplements in a course setting. The authors have used them in calculus and upper-level classes as homework assignments and for in-class discussions. Students have been asked to justify the one they have chosen, offering opportunities for noncomputational critical thinking, speaking, and writing about mathematics.
Table of Contents:
Part 1: Precalculus 1. Integers: Rational Numbers 2. Circle: Center and Radius 3. 3 − √7: 3 1√7 4. y = logb x: x = by 5. Even-Degree Polynomial: Horizontal Line Test 6. Nonorigin Point in the Plane: Polar Coordinates (r, θ ) with 0 ≤ θ < 2π 7. am: am−n 8. x = 0: log10 x 9. Degree of a Polynomial: Number of Real Roots 10. ex: ln x 11. Complex Numbers: Real Numbers 12. x2+1 : x = 1 13. y = arctan(x) + 3: y = arctan(x) 14. √ab: √a√b 15. Positive Integers: Prime Numbers and Composite Numbers 16. Polynomial Function: Asymptote 17. x2 = x: x = 1 18. Circle: Ellipse 19. (A ∩ B) ∪ (A ∩ C): A ∪ (B ∩ C) 20. log2 x: ln x 21. tan x: cot x 22. Function: Symmetric about the x-axis Part 2: Calculus 23. Rolle’s Theorem: Mean Value Theorem 24. y = cos x: y − 1 = √3− 2 (x − π ) 25. Riemann Sum: Definite Integral 26. Critical Point: Local Extremum 27. limx→π sin x: sin x 28. Marginal Cost: Total Cost 29. limx→0 cos x: L’Hôpital’s Rule 30. Differentiable : Continuous 31. Maclaurin Series : Taylor Series 32. Concave-Up Function : Minimum 33. Definite Integral : Area 34. Factorials : Polynomials 35. Addition : Arithmetic Sequence 36. 1: ln x 37. Velocity: Position 38. Taylor Polynomials: Smooth Function 39. Chain Rule: Integration by u-substitution 40. Improper Integral: Definite Integrals 41. Everywhere-Zero Derivative: Differentiable Function 42. f (xx) −f (a): f ′(a) 43. ∞ 3 2 n: 3 44. Intermediate Value Theorem: |x| on the Interval [−2, 1] Part 3: Postcalculus 45. Dot Product: Orthogonal Vectors 46. □□2f 47. Gradient: Level Curve 48. □□2f □□2f − а □□2f 49. Three Distinct Points: Plane in R3 50. Gradient: Directional Derivative 51. Concavity: Euler’s Method 52. Separation of Variables: Homogeneous Differential Equation 53. Switching Two Rows: Determinant 54. Matrix: Reduced Row Echelon Form 55. Eigenvalue: Eigenvector 56. Determinant: Rank 57. Matrix Multiplication: Square Matrices 58. Permutations: Determinant 59. Event: Sample Space 60. Successive, Independent Coin Flips: Probability of Heads 61. Cumulative Distribution Function: Probability Density Function 62. Geometric Distribution: Exponential Distribution 63. Eulerian Path: Graph 64. Tree: Cycle 65. Bipartite Graph: Three-Colorable 66. Planar Graph: K5 Subgraph
About the Author :
Katie Ballentine is a software engineer at Guild Education. Previously, she was Director of Communications at Mathematical Sciences Publishers (MSP), and before that, she was a mathematical copy editor at MSP and Mathematical Reviews. She has taught a variety of undergraduate math classes.
Michael A. Jones is the Managing Editor and an Associate Editor at the American Mathematical Society’s Mathematical Reviews. He was a faculty member for 14 years, including as an Associate Professor in mathematics at Montclair State University, New Jersey. He served as the editor of the Mathematical Association of America’s Mathematics Magazine from 2014 to 2019. In the summers, he teaches high school students through the University of Michigan’s Michigan Mathematics and Science Scholars program. He is the author of approximately 70 research and expository articles in mathematics and its application to the social sciences.
Lon Mitchell is currently a professor at Eastern Michigan University, was previously an editor for the American Mathematical Society, and has taught for multiple gifted high-school enrichment programs, including for the University of Michigan.
Jennifer M. Wilson is an Associate Professor of Mathematics at The New School, where she has been a faculty member since 2003. She has taught most of the standard courses in the undergraduate mathematics curriculum and has published research in a variety of different journals, including Mathematics of Social Sciences, Mathematical Methods of Operations Research, the College Mathematics Journal, and Mathematics Magazine.
