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Discovering Mathematics: The Art of Investigation

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Mathematics is the act of investigating what happens when you set down a set of objects, a set of rules on what you can do with the objects and then do things following those rules. Sometimes we put forward educated guesses as to what our result will be and at other times we simply just do what we can do to see what happens.
Discovering mathematics is nothing more than doing these things and in this book, Gardiner starts with some simple games that are easy to analyze. In the next section, a generalized three-digit number is selected, the three digits reversed and the process is repeated. List of self-referential sentences such as

"This sentence uses thirty symbols."

are analyzed next. Further topics involve geometric figures, an extensive section involving more flipping of the digits in the positions of a number, simple recurrence relations, classic jug and postage stamp problems, solving linear equations and coinage problems. Large numbers of additional exercises are given at the end of the chapters and hints followed by solutions are also included.
This book should be considered more a primer on mathematics than a book on mathematics. The problems are more in the area of mathematically based puzzles rather than what we would consider "pure" math problems. Nevertheless, it is a good source of quick problems for classes in mathematics for education majors or classes in algebra near or at the level of precalculus.