Reviews (4)

Mind-blowingly original, side-splittingly funny
This is not a book for mathematical beginners, even though it starts from literally nothing.But readers who have learned enough traditional math to understand the point of set theory and who have a solid grasp of the real number system are in for a wild ride, and will never look at numbers, or games, in the same way again.

Conway is the most original mathematician on the planet, as well as a remarkably witty and vivid writer, who combines wordplay and logic better than anyone since Lewis Carroll. The book is far too densely packed to summarize in a short review. All I can say is that it's practically inexhaustible; like all good math books, what you get out is proportional to the effort you make while reading it, but the amount of effort it will repay is a hundred times as much as for an ordinary book.

This is an all-time classic, a "desert island book".Even though this new edition differs from the old one in very minor ways, I bought it immediately because my 1978 copy was falling apart from extreme overuse.(My other "desert island math book" is Cohen's "Set Theory and the Continuum Hypothesis".)

Math geek heaven
Boy, you wanna talk about your _cool_ books. I read this one twenty years ago and never quite got over it. Georg Cantor sure opened a can of worms with all that infinity stuff.

John Horton Conway is probably best known as the creator/discoverer of the computer game called "Life," with which he re-founded the entire field of cellular automata. What he does in this book is the _other_ thing he's best known for: he shows how to construct the "surreal numbers" (they were actually named by Donald Knuth).

Conway's method employs something like Dedekind cuts (the objects Richard Dedekind used to construct the real numbers from the rationals), but more general and much more powerful. Conway starts with the empty set and proceeds to construct the entire system of surreals, conjuring them forth from the void using a handful of recursive rules.

The idea is that we imagine numbers created on successive "days". On the first day, there's 0; on the next, -1 and +1; on the next, 2, 1/2, -1/2, and -2; on the next, 3, 3/4, 1/4, -1/4, -3/4, and -3; and so on. In the first countably-infinite round, we get all the numbers that can be written as a fraction whose denominator is a power of two (including, obviously, all the whole numbers). We can get as close to any other real number as we like, but they haven't actually been created yet at this point.

But we're just getting started. Once we get out past the first infinity, things really get weird. By the time we're through, which technically is "never," Conway's method has generated not only all the real numbers but way, way, way more besides (including more infinities than you've ever dreamed of). His system is so powerful that it includes the "hyperreal" numbers (infinitesimals and such) that emerge (by a very different route, of course) from Abraham Robinson's nonstandard analysis as a trivial special case.

So there's a lot here to get your mind around, and it's a lot of fun for readers who like to watch numbers being created out of nothing. But wait -- there's more.

See, the _full_ title of the book includes not only "numbers" but also "games". And that's the rest of the story. Conway noticed that in the board game of Go, there were certain patterns in the endgames such that each "game" looked like it could be constructed out of smaller "games". It turns out that something similar is true of all games that have certain properties, and that his surreal numbers tie into such games very nicely; "numbers" (and their generalizations) represent strategies in those games. So in the remainder of the book Conway spells this stuff out and revolutionizes the subject of game theory while he's at it.

Well, there must be maybe two or three people in the world to whom this all sounds very cool and yet who haven't already heard of this book. To you I say: read it before you die, and see how God created math.

A very dense collection of original ideas
We all think we know numbers, and yet every once in awhile something comes along that makes us realize that we actually know very little. I am not talking about facts such as whether a specific large number is prime, but about the fundamental definition of what a number is. The appearance of the surreal numbers is one of those mathematical equivalents of a whack on the side of the head. Suddenly, numbers are defined as the strengths of positions in certain games, something that is at first strange, but it turns out that the class of objects defined this way includes the real and ordinal numbers. It certainly is different, and I had to read the first thirty pages of the book three times before I felt that I truly grasped the concepts behind the definition of the surreal numbers.
From that things move more smoothly. As I read through the book, it was easy to get the impression that most of life can be described as a game, where our day-to-day status in the community can be described as a dynamic set of surreal numbers. I often wondered if that may be an effective approach for artificial intelligence work, as it certainly seems that surreal numbers can be used to model almost any dynamic situation. Furthermore, effective game playing is nothing more than effective decision making.
There are many significant ideas in the book, at times you stop and start mentally jumping through different scenarios, as in "What would be the change if this rule is added, dropped or altered?" It seems that if you took that approach, several lifetimes could be spent in exploring all the possibilities. I have read many books and this one is most likely the densest carrier of new ideas that I have ever encountered.

Reviews (4)

Improvement!
These new editions have many new and interesting stuff that is not included in the original outdated series. It contains many fresh ideas that the authors recently discovered including those old ones. For old ones the original volume has more to say...

Geniuses and Games
This book is dazzling.It can be pretty tough going but it is
well worth the effort.You can always tell the work of a genius because it illuminates the landscape and shows us things we have never seen before.I design games for a living and this book rocks!Hackenbush, Nimbers, games with 1/2 move advantage.Well illustrated.ONLY PROBLEM: Where are volumes 2-4?

Isbn: 156881142X
Sales Rank: 205314
Subjects:  1. Game Theory    2. General    3. Mathematical recreations    4. Mathematics    5. Recreations & Games    6. Science/Mathematics   

Reviews (2)

Great reference for one person puzzles and games
The new edition of these classic volumes has been completely reorganized, and this volume now contains mostly one person games or puzzles, such as peg solitaire, Soma, Rubik's Cube, mechanical wire and string puzzles, sliding block puzzles, magic squares, and life.The book is very readable and requires no mathematical background.However, this is no lightweight watered-down book and some sections of the books could take you months to understand completely (try the SOMA map or century puzzle map that appears in the Extras).Fortunately you can just skip over these parts if you don't want to dig down to this level of detail.

I have only looked briefly at the other volumes, but I believe this volume "stands on it's own" more so than volumes 2 & 3.Be warned, however, that there are several concepts (such as "nim addition" that you will need the previous volumes to understand).

Conway's game of Life is the subject of the last chapter, perhaps the most interesting chapter in the book, and that which has probably been most changed since the last edition.Still, they could easily have expanded this chapter into a whole volume, and looking at the internet it is already out of date.

Beware that the figures on the covers of these volumes DO NOT necessarily correspond to what is inside.For example, Volume 3 shows peg solitaire on the cover but the subject itself is all in Volume 4!

Reviews (3)

Fascinating
I don't have read all the book, but I read most of the scientific papers it is composed by. I think they are very interesting and puzzling, on the border line between serious mathematics (game theory and all this stuff) and "recreational math" (like the angel problem). It would be a good read also for people interested in computer games.

great book
This book is full of beatiful work.Every section is an investigation into some combinatorial game, or some idea in combinatorial game theory.Most of the material is clearly presented and all should be accessible toundergrads, but be warned: this is not simple stuff.But, as we all know,beautiful mathematics isn't always simple.The book also includes asection with 52 unsolved problems, which should be of considerable interestto the curious.

Isbn: 0521808324
Sales Rank: 577610
Subjects:  1. Combinatorial Analysis    2. Combinatorics    3. Congresses    4. Game Theory    5. Mathematics    6. Science/Mathematics    7. Combinatorics & graph theory    8. Mathematics / General   

Reviews (3)

The only book about Dots and Boxes to my knoweledge
This book is one of a kind, and will show you that the game of Dots and Boxes is far from trivial, in spite of the common idea. It will also show you some very high-quality technical stuff about this game.The main issue is that the book is born as a collection of problems, and the try to turn it into a full-fledged book wasn't completely successful. The explanations are lacking in clarity and the result is that you have a very good series of positions and problems, but not the knoweledges for solving them

Incomplete
Although the subject matter is enticing and the introduction thorough, the text proper falls short.The examples come with minimal, if any, explanation, leaving it to the reader to actually make conjectures and prove lemmas.This book seems to have been writtem more as a private notebook for someone who already understands the game rather than an explanation for neophytes.

Reviews (3)

A Minute to Learn: A Lifetime to Master
Hex is a deterministic, zero-sum, abstract board game. If you know what this means, and appreciate such games, then you will find the game irresistible. The rules are so simple that you can "see" far down the strategy tree. Draws are impossible. Hex was invented independently by two eminent mathematicians, Piet Hein and Nobel Prize winner John Nash (the latter of _A Beautiful Mind_ fame). The feel of competition that this game provides is intense; one player compared it to a "knife fight in a phone booth." The game demands your best efforts, and rewards them.
If you are talented mathematically, there are chapters which deal with the game in a way that appeals. If not, you will still love to study how the book progresses from the simplest templates to tactics to overall strategy. Annotated games are given, as well as quizzes and problems. Game of expert play are taken from the internet. Hex programs are dicussed. There is a rich treatment of the variants and offshoots of the basic game, although perhaps basic 11 x 11 hex is probably still the best of the bunch. The author creates a rich vocabulary to describe the different aspects of the game, and while the reader may have to slow his reading occasionally and ponder, he will find everything in its proper order and will find that everything makes sense. I found a useful colection of blank boards of different sizes at the back of the book which I photocopied to make studies of the games I have now in progress on the net, and thereby explore the different possible avenues of play by using pencil-and-paper diagrams. Anyone who enjoys abstract games such as chess and go will be cheating himself by not exploring the richness of this book, and the richness and challenge of this game. There are a few typos here and there, but relatively few, and not very distracting.

The "Hex" Bible
Being hex rules so simple, there are only 2 possibilities. Either you never heard about hex or you know how to play it. In the first case there are good odds that you can like this book,especially if you like games such as chess or checkers. But in the second, this book is a must. There is everything about this wondeful game, from strategy to historic notes, from variants to computer-play and algorithms, from sample games to a great reference section.
So this is THE hex book.

Isbn: 082180166X
Sales Rank: 3090783
Subjects:  1. Combinatorial Analysis    2. Congresses    3. Game theory    4. Science/Mathematics   

Isbn: 0471127388
Sales Rank: 838405
Subjects:  1. Algebra    2. Algebra - Elementary    3. Algebra - General    4. Algebra - Intermediate    5. Algebra - Linear    6. Mathematics    7. Science/Mathematics    8. Mathematics / Algebra / Linear