Martin Gardner wrote his "Mathematical Games" in *Scientific American* in the period (1956-1981). In these columns he discussed in an enteraining way some mathematical game or puzzle or another mathematical topic in a playful way. These columns were bundled, some 20 columns per book, and extended with further comments and solutions and were marketed by diverse publishers. Some of there have been revised and reprinted.

Since 2008 the Mathematical Association of America and Cambridge University Press have set up a 15 volume series: *The New Martin Gardner Mathematical Library*. The idea is to bring all these collections together in the same series and reprint updated versions of the old books. After Martin Gardner passed away in 2010, a bunch of editors have taken over the task of updating and commenting on the many different topics.

This book is the new edition of *Unexpected Hangings, and Other Mathematical Diversions* originally published by Simon & Schuster (1968). This is a revision and extension of that book (more variations of the games and better solutions to the puzzles became available in the mean time, and the reference list needed an update).

In this case the updates were written by Martin Gardner, Peter Renz, Greg Frederickson, and Erica Flapan. With this book, we now have volume 4 in the new series. A review of the first two volumes is available in the EMS database: Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi (vol.1), Origami, Eleusis, and the Soma Cube (vol.2).

Martin Gardner's prolific work in which he developed as an unrivaled ambassador of playful and recreational mathematics, games, puzzles, and magic, does not really need an introduction. All these topics are also represented by the twenty chapters in this volume. The format is as in the other volumes of the series. Most often, a problem is stated, some variations and complications or handicaps are added, which can be formulated as challenges for the reader. At the end of the chapters, usually answers are provided and updates are added in the form of an afterword. Obviously also the list of references for further reading is updated and extended. Noteworthy is that many graphics are added for better understanding. Sometimes these are not fully in correspondence with the text. For example on page 38, the text refers to dotted lines which are full lines in the plot and on page 128 the text says that rows are numbered from top to bottom, while the numbering is reverted in the graphic. These flaws however are not really hampering the readability.

This is not the place and the space to go through all the subjects but there is logic and paradoxes, knot theory, probability, artificial intelligence, number theory, card tricks, plane tilings, topology and so much more, but always fun. Never dull mathematical derivations with an overload of formulas. To give an idea about the contents we list the titles of the chapters

- The Paradox of the Unexpected Hanging
- Knots and Borromean Rings
- The Transcendental Number e
- Geometric Dissections
- Scarne on Gambling
- The Church of the Fourth Dimension
- Eight Problems
- A Matchbox Game-Learning Machine
- Spirals
- Rotations and Reflections
- Peg Solitaire
- Flatlands
- Chicago Magic Convention
- Tests of Divisibility
- Nine Problems
- The Eight Queens and Other Chessboard Diversions
- A Loop of String
- Curves of Constant Width
- Rep-Tiles: Replicating Figures on the Plane
- Thirty-Six Catch Questions

The Martin Gardner adepts will eagerly acquire this volume and start longing for the next volume to come out as they want their series to be complete. But if you are not such a groupie or fan, just this volume is a guarantee for several free afternoons of brain teasing amusement and mathematical experimentation with cards, strings, paper, rings, leather braiding, or boardgaming with friends.