This book is a fun exploration of "shoelace mathematics". The author first classifies various lacing patterns according to various criteria, and then derives formulas and proofs answering questions like "What are the shortest laces you need to lace a shoe, and what pattern do you lace it in" and "What is the strongest lacing pattern".
The book has fairly simple prerequisites: it uses algebra, uses the combinatorial formula and series of sums, and, in the section on the strongest lacings, some calculus. Most of the math can be understood by working it out on paper, but there are some questions that might occur to the reader that a graphing calculator might be useful for. Or a program like Microsoft Math.
The best part of the book is the attitude it teaches: You do mathematics by starting with a very simple question like "What is the shortest way to lace a pair of shoes" and investigate it, then go on to other questions that occur to you, and on and on, until you're answering all kinds of related questions. It is an exploration. It also teaches how you can do this: You start by breaking off smaller problems that you can answer, using models that are simpler than what you're after, and after searching for and finding solutions to the simpler problem you can start answering related questions, and then more difficult questions. There are plenty of shoelace math questions that might occur to the reader that he could go on to investigate on his own.
The book also shows that mathematics needn't be totally dry. It entertains with photos of real shoes laced in various ways, has some Peanuts and Dilbert and other cartoons, discusses related problems (like the Traveling Salesman problem), and even some history of shoelacing.
The author also wrote a book on the mathematics of juggling, called (of course) "The Mathematics of Juggling". That is also a terrific book showing how you can investigate the mathematics of a problem - I think it is more difficult mathematics, however.