There is a one-to-one correspondence between the ducks on the left and
the eggs on the right. Actually, there are *many* one-to-one
correspondences... *How* many?

*Discrete Mathematics with Ducks* (first edition AK Peters/CRC Press 2012, second edition in press, appearing in late 2018) is a
textbook intended for a sophomore-level course in discrete mathematics. It does
not have any mathematical prerequisites and it does not assume any prior
exposure to proofwriting. Still, the focus is firmly on actual discrete
mathematics content rather than general mathematical background. *Discrete
Mathematics with Ducks *is particularly amenable to use by faculty who like
a discovery-based approach; in-class activities are supplied for every topic,
and about half of the topics are presented so that students' first exposure to
the material is via explorations in class. For this reason, the book is also
amenable to self-study. However, the mathematical presentation would also work
with more common teaching styles. There are lots and lots of homework problems
as well, including very straightforward problems at the end of explanatory
sections, the usual sorts of homework problems at the end of each chapter, and
50 additional problems at the end of each theme. Each chapter also has 10 homework-level problems for which solutions are provided. For those instructors
wishing to try collaborative learning, a practical guide is provided.

The curriculum complies with SIGCSE guidelines (see outline below), and both practical applications and light-hearted examples are given throughout. The material is designed to be manageable for those students who are stymied by a first introduction to proof, yet with scalable exercises to challenge those students who otherwise would consider the class a breeze. Each chapter is intended to take one week of class time, and contains a mixture of discovery activities (for self-study or work in class), expository text, in-class exercises, and homework problems. Additionally, bonus material is included for enrichment or fast-paced classes, and all chapters contain guides to further study and suggestions for instructor use.

One can purchase *Discrete Mathematics with Ducks* from CRC or through
amazon.com
(amazon.co.uk,
amazon.ca,
amazon.jp),
Barnes&Noble,
Foyles
(UK), Book
Depository (UK), Saxo
(Denmark), Angus&Robertson
(Australia), Fishpond
(New Zealand), Flipkart
(India), IBS
(Italy), and there are certainly other sources not yet noted.

Reviews of *Discrete Mathematics with Ducks*: on
amazon.com and MAA
Reviews.

**Frontmatter:** Preface for teachers, Preface for learners,
Tips for Reading Mathematics, Problem-Solving Prompts, Tips for Writing
Mathematics

**Theme 1: The Basics**

**Chapter 1:** Counting and Proofs (sum and product and pigeonhole
principles; direct proof and disproof)

**Chapter 2:** Sets and Logic (notation, double-inclusion, Venn
diagrams, statements, quantifiers, truth tables, contradiction, contraposition;
bonus: truth-teller puzzles)

**Chapter 3:** Graphs and Functions (definitions, examples,
one-to-one, onto, graph isomorphism, graph operations; bonuses: party tricks,
characteristic function)

**Chapter 4:** Induction (the proof technique, examples from sets,
counting, graphs, sums; bonus: minimal criminal)

**Chapter 5:** Algorithms with Ciphers (algorithm definitions and
examples, modular arithmetic, equivalence relations, shift and vigenere
ciphers; bonuses: depth- and breadth-first search, pigeonhole principle with
divisibility)

**Theme 2: Combinatorics**

**Chapter 6:** Binomial Coefficients and Pascal's Triangle (choice
numbers, overcounting, permutations, combinatorial proof; bonuses: bubblesort,
Mastermind analysis)

**Chapter 7:** Balls and Boxes and PIE (combinatorial problem
types and solutions, inclusion/exclusion; bonus: linear and integer
programming)

**Chapter 8:** Recurrence (Fibonaccis, induction connections,
finite differences, characteristic equation; bonus: writing recurrences from
situations)

**Chapter 9:** Counting and Geometry (hyperplane arrangments;
bonus: geometric gems)

**Theme 3: Graph Theory **

**Chapter 10:** Trees (spanning trees, spanning tree algorithms,
binary trees, matchings, backtracking; bonus: branch-and-bound)

**Chapter 11:** Euler’s Formula (proof, applications; bonus:
topological graph theory)

**Chapter 12:** Traversals (Euler circuits, Hamilton circuits,
TSP, Dijkstra's algorithm; bonuses: RNA chains, network flows, Hamiltonian
criteria)

**Chapter 13:** Coloring (vertex coloring, edge coloring,
applications; bonus: the four color theorem)

**Supplemental/Optional Material**

**Chapter 14:** Probability and Expectation (probability terms and
examples, expected value, conditional probability; bonus: the probabilistic
method)

**Chapter 15:** Cardinality (un/countability, bijections between
infinite sets, continuum hypothesis; bonus: the Schröder-Bernstein theorem)

**Chapter 16:**Number Theory (Euler phi function, gcd/lcm, solving congruence equations, mediants; bonus: RSA encryption/decryption)

**Chapter 17:**Computational Complexity (computing and comparing runtime functions; complexity classes)

**Backmatter:** List of Symbols, Solutions to
straightforward problems, Solutions to homework-level problems, Reference List, Index

Full Table of Contents (first edition) (.pdf)

If you are interested in an examination copy of *Discrete Mathematics
with Ducks*, please use the "Request Evaluation Copy" button at the publisher's page
to request one.

Support Files and Web Resources
mentioned in *Discrete Mathematics with Ducks*

**Frequently Asked Question:** Why did you title the book
*Discrete Mathematics with Ducks*?

**Answer:** Many discrete mathematics textbooks have titles
such as *Discrete Mathematics with Graph Theory* or *Discrete
Mathematics with Applications*. I think these titles are ridiculous---it
would not make sense to have a discrete mathematics text without applications
or without graph theory. My title is poking a bit of fun at this situation.
Besides, I never miss an opportunity to be silly! It's important to treat
serious mathematics with levity when possible.