# The Magic Garden of George B and Other Logic Puzzles

This book was published in 2007 by *Polymetrica*, an Italian open access book publisher. However, since it does not seem to exist anymore, it is now made available as a World Scientific publication in 2015.

In fact it contains two books. The first is a collection of logic and mathematical puzzles of the type that made Smullyan popular among his fans since he published his first collection *What is the name of this book?* in 1978. The promotion by Martin Gardner made it an instant success and later a dozen similar collections have appeared. Smullyan is a mathematician with a PhD from Princeton (1959) that he prepared under Alonzo Church on Gödelian incompleteness theorems in formal systems. But he is also a gifted piano player, he worked as a magician and he published on Taoism, philosophy and religion. His recreational books may not be as diverse as Martin Gardner's but he has an outstanding ability to bring his logical puzzles and riddles in many variations and in a most entertaining way. Some are easy, but some are quite challenging, even for experienced puzzlers.

The first book in this collection has 12 chapters where first, if needed, a general setting is explained. For example: Teresa, Thelma, Leila, and Lenore are four sisters the first two always tell the Truth, the other two always Lie. Then there follows a list of problems to solve. For example: If you meet one of them, but don't know which, what statement should she make to convince you that she is Lenore? Each chapter ends with the solutions and the arguments used. Most puzzles are logic, but some of them involve some algebra with integers or even some probability. Truthful to his nature, Smullyan cannot refrain himself from occasionally including a joke here and there, but there are not too many.

The second book is an introduction to Boolean algebra (the George B of the title is of course George Boole). It starts out in the previous recreational style, but it soon mover to a more formal treatment. So this part is somewhat more demanding for the layman with proper formulas and truth tables, and even definitions and theorems. It starts with a first 'grand problem'. In George's magic garden grow flowers that are either red or blue for a whole day, but they may change their color from day to day. If you pick any three flowers on any day, then if the first two are both blue, the third one is red and if the first two are red, the third is blue. Moreover, for any two distinct flowers, there is at least one day on which their color will differ. If you know that there are between 200 and 500 flowers, how many are there exactly? The puzzle is not trivial, but there is exactly one solution. The problem is solved but only after some exercises with set theory, unions, intersections, complements, etc. which allow to tear it down in several solvable subproblems. Propositional logic and formal Boolean algebra's are the next mathematical fortresses to conquer. The ultimate 'grand problem' is to prove that one has a Boolean algebra once a set of axioms is satisfied.

Hence what started as some fun problems, that in principle are solvable by anyone, the second book takes a turn towards a much more formal approach and mathematical ingredients which gives a proper introduction to propositional logic and Boolean algebra. The hope is of course that readers attracted by the first part will engage also into the second book if they want to learn a systematic approach to solve the logical problems. I doubt that they will reach the end unless they are strongly motivated since eventually indeed all the 'fun' elements have evaporated and turned into mathematics which, unfortunately is considered the opposite of fun by many. So this is a most remarkable attempt to join the extremes in one volume and hopefully some will be convinced. Of course for the professional, there is not much new in the mathematics, but he or she will certainly enjoy the puzzles. For the student who has to master the mathematics, this may turn out to be a very enjoyable way to get acquainted with the subject, guided by Smullyan, who is the grand master of this kind of entertainment. They will not be deceived. As an introduction to formal logic, his other book *Logical labyrinths* (A K Peters, 2007) is better and more elaborated. It does not stop after the propositional logic but also includes a further treatment of first order logic.

**Submitted by Adhemar Bultheel |

**2 / Jun / 2015