Dennis Shasha has written a superb sequel to his first book of mathematical puzzles, The Puzzling Adventures of Dr. Ecco, which appeared in 1988 . This second book is a reprint of a 1992 book that was originally published as *Codes, Puzzles, and Conspiracy*. The third book in the series, published in 2002, is *Dr. Ecco's Cyberpuzzles: 36 Puzzles for Hackers and Other Mathematical Detectives*. He also writes a regular puzzle column in Dr. Dobb's Journal called "Dr. Ecco Omniheurist Puzzle Column." Shasha is Professor of Computer Science at the Courant Institute of Mathematical Sciences at New York University.

This book, like the first one, stars the inimitable Dr. Jacob Ecco, a mathematical prodigy, and his cohorts Dr. Evangeline Goode, a Princeton philosopher, and Professor Justin Scarlet, topologist at the Mathematics Institute in New York. Dr. Ecco is a world famous omniheurist as are his two friends. An omniheurist is a person who can solve problems of all types. We are told that "the science of omniheuristics applies a mathematical approach to the solving of problems. Omniheuristics is more a method, a way of seeing things, than a specific field of study. Specialists may supply critical data, but the generalist omniheurist through insight and reason alone, finds the solution."

As in the first book, the three friends go about solving various problems brought to them by company executives, military generals, and casino owners. The book opens with the disappearance of Ecco. It appears that Benjamin Baskerhound, also a Princeton mathematician, but a rogue genius who uses his love of puzzles to commit crimes, has kidnapped Ecco. Goode and Scarlet are hot on his trail and their adventures take them all over the world — from New York to Brazil to Ireland to Thailand to Japan and so on. In Ecco's absence, Dr. Smartee, a graduate of Eton and Christ Church, Oxford, has earned the title of best omniheurist. But is he legitimate?

The story also serves as a telling non-partisan commentary on the current state of the nation. The villains in the story are not necessarily who you think they are, but they certainly correspond correctly to the villains in our continuously unfolding national saga. The bad guys seek to impose a totalitarian society based on extensive surveillance. People are so filled with fear of attack that they are willing to give away their freedom in the mistaken belief that it will bring them security. Boasts by one of the villains such as "within a year, I will have imposed a truth so total that no sane person will consider questioning it. The few doubters will be labeled deviants" sound chillingly real. Fortunately in the story "the nation at large was not in tune with such sentiments." One hopes that reality follows fiction in this instance. Saying anything more would give away the surprises in the story.

Now let's talk about the mathematics in the book. There are forty-five puzzles woven into the story. Complete solutions and further readings are given at the end of the book. Broadly speaking the puzzles can be categorized thus: ten on codes, four betting games, four voting puzzles, one puzzle involving modular arithmetic, two on graph theory, one on the pigeon hole principle, one on combinatorial designs, one on Thue systems, one on protocols in computer systems, one on toxicology, two logic puzzles, two geometry puzzles, and the rest fall in the broad area of discrete math.

Here is an example of a probability problem than is perfect for any course that has a chapter on probability. A single roller plays against the casino. The roller can bet either 'Hi' corresponding to dice totals of 8 through 12 or 'Lo' corresponding to dice totals of 2 through 6. If the roller rolls a number corresponding to his bet, he wins an amount 25 percent more than he placed. Assuming fair dice, what would the odds be?

Here is a puzzle based on protocols in computer systems that are used to verify the identity of computerized agents. The sand analogy is reminiscent of Archimedes famous work on large numbers *The Sand Reckoner*. The Amazing Sand Counter claims that if you put sand into a bucket he knows at a glance how many grains there are. But he won't tell you the number. So why would anyone believe him? That is precisely the question. Can you devise a test by which the Sand Counter can prove his skill without telling you anything that you don't already know?

A distinct feature of the puzzles is that most of them have a follow-up puzzle with a slight change in the settings. The second problem is more difficult than the first. This feature is especially nice because it encourages further thought on the subject. However, as the author says, one doesn't need any mathematical training to solve and appreciate these puzzles. All one needs is a love of puzzles.

The book has some unusual pieces of information. For example, we all know about the deductive and inductive style of inference. But did you know there is another form of inference between deduction and induction called abduction? This is an interesting snippet in keeping with the kidnapping theme of the book. There is a reference to an 1878 paper by Charles Sanders Peirce. Specifically, we all know the following as a classic syllogism:

All the beans from this bag are white.
These beans are from this bag.

Therefore, these beans are white.

Then we have inductive logic:

These beans are from this bag.
These beans are white.

Guess that all the beans from this bag are white.

Reasoning using abduction or retroduction is as follows:

All the beans from this bag are white.
These beans are white.

Guess that these beans are from this bag.

This type of reasoning is not like an argument from population to sample (deduction) or an argument from sample to population (induction). It is a form of probable argument different from both deduction and induction. (See http://plato.stanford.edu/entries/peirce/.)

Evidently the first book in the series was used by Andy Liu of the University of Alberta as the basis of a successful college course in mathematics similar to the usual mathematics for liberal arts course offered by most colleges. You can find his review at amazon.com. The University of Alberta published *Professor Scarlet's Notebook*, a companion piece that Liu and his coauthors wrote for anyone interested in pursuing the mathematics behind the puzzles. It would be nice if someone did the same for the collection of puzzles in this book.

The three books chronicling the adventures of Dr. Ecco are must-haves for university libraries as well as county public libraries. Puzzle enthusiasts will love this book. Middle school, high school, and college level math and computer science club advisors will find much material to motivate students and engage their curiosity.

Sandra Kingan (srkingan@psu.edu) is assistant professor of mathematics at the Pennsylvania State University, Capital College. Her research area is combinatorics. She is also the current president of the Harrisburg Chapter of the American Statistical Association. Her broad mathematical interests became broader as a result of going through the puzzles in this book.