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.
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. The puzzzles can be used in calculus and upper-level classes as homework assignments and for in-class discussions.
Part 1: Precalculus
1. Integers: Rational Numbers
2. Circle: Center and
Radius
3. 3 7: 3 17
4. y = logb x: x = by
5. Even-Degree Polynomial:
Horizontal Line Test
6. Nonorigin Point in the Plane: Polar Coordinates (r,
) with 0 < 2
7. am: amn
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: ab
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. Rolles Theorem: Mean Value Theorem
24. y = cos
x: y 1 = 3 2 (x )
25. Riemann Sum: Definite Integral
26. Critical
Point: Local Extremum
27. limx sin x: sin x
28. Marginal Cost: Total Cost
29. limx0 cos x: LHôpitals 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 2f
49. Three Distinct Points: Plane in R3
50. Gradient:
Directional Derivative
51. Concavity: Eulers 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
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 an associate editor at the American Mathematical Societys Mathematical Reviews. He was a faculty member for 14 years, including as an associate professor in mathematics at Montclair State University, NJ. He served as editor of the Mathematical Association of Americas Mathematics Magazine from 2014 to 2019. In the summers, he teaches high school students through the University of Michigans Michigan Mathematics and Science Scholars program.
Lon Mitchell is currently an associate professor of mathematics 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.
