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 (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.

Reviews (5)

Fascinating Background Material to Knuth's Typesetting Work
This book won't teach you TeX or Metafont. It might not even teach you all that much about particular algorithms (although Chapter 3 is one of the most detailed explanations of TeX's linebreaking algorithms published anywhere). Instead, this book offers a look behind the scenes.

Instead of beholding TeX and Metafont in their almost final versions, as published in _TeX: The Program_ and _Metafont: The Program_, respectively, you see them grow from the first design studies (when Knuth thought of TeX as a program for two grad students to write over a summer) to where they are today. You see how the collaboration between Knuth and Zapf on the Euler fonts worked, and you get another glance at many facets of Knuth's mind (And a beautiful mind it is indeed, even though it is entirely sane).

If you have any deeper interest in TeX and Metafont, this book is well worth the money.

MASTERFUL
EXCELLENT book.... I cannot rate this one high enough.... at firstI thought it might have been expensive but it is NOT... the price is well WORTH it for what you get, Knuth is a master!

Editorial Review

Reviews (5)

Knuth is Analysis of Algorithms
Befor Donald Ervin Knuth, there was no such thing as the Analysis of Algorithms.He is a visionary in this field, and these selected papers are a testomony to his greatness in this field.

Not as Approachable as Knuth's Other Writings
To a reader interested in the analysis of algorithms, this bookdoubtlessly contains a number of very thorough analyses, with someinteresting correspondence about the origin of complexity O()-notation andthe term "NP-hard" thrown in.

As a reader more interested inKnuth's work in general, I must admit that, despite having advanced degreesin CS and a quite decent math education, I found myself unable to followthe heavy mathematics employed, and ultimately, the problems analyzed werenot interesting enough to me to make me spend the effort to follow it. Oneof the chapters was titled "A Trivial Algorithm Whose AnalysisIsn't", and this probably sums up why I'm not entirely enthusiasticabout this book and about the field of algorithm analysis ingeneral.

Knuth wouldn't be Knuth if he didn't throw some lighter materialin as well, and chapter 2, "The Dangers of Computer ScienceTheory", was quite amusing.

Reviews (24)

A pourry of combinatorics
I want to start saying that this is a book designed for Engineers, not for Mathematicians. It focuses on the tecniques, not on the arguments. This is not a book about combinatorics, it is a wide raging introduction (it lacks on definitions, and his proofs are a lot far away from mathematical ones). The Enegineers can use this book as a good reference. The Mathematicians can improve their lateral thinking, for them (well: us) it is book about problem-solving strategies.
I will never use this book as a textbook for a graduate/undergraduate course, it can be helpful if used with another book about combinatorics: when you study a combinatorial object, you can read from this book the techniques it involves.
The exercises are extremely exciting, when I read this book I spent a lot of time about its exercises (proportion read:solve = 1:3), and they led me to interesting results.

Steep learning curve, the definitive prerequisite for TAOCP.
Why I got this book:
It's a great feeling to know how computers work, when I decided that I want to make a career and a life out of computers, as its truly a passion for me, I delved deeper, discovering the true beauty in the Science part of Computer Science, so I decided to get Donald Knuth' "The Art of Computer Programming" - to describe that seminal, huge work, it's like biting more than you can chew while trying to drink from a fire hose, moreover, the technical and mathematical prerequisites for the work are sometimes too demanding, they require a huge amount of experience with discrete mathematics, although I had some lectures and read some books, none came close "Concrete Mathematics", it covers, from ground up (though with a dangerously steep learning curve) a lot of discrete mathematics topics, it is by far the most extensive work I've read about Sums and really teaches the algorithmic problem solving thinking skill the authors preach so much about, with small amusing comments written by actual students of this course, a comfortable format, and very good writing skills, you can feel these guys are great professors who enjoy this material and are passionate about teaching it.

Recommended, though some better, less steep, introductionary text books are probably out there.

Enjoy.

Only one problem with this textbook
Basically, I like this textbook. The material is interesting, the way the authors presented the material is inspiring, and they provided a lot of jokes to make even studying for exams not that boring. But there is one big problem which made me decided to rate this book only 3 stars instead of 5 stars: the authors like to use non-standard notations. For example: m\n means "m>0 and n=mk for some integer k". One of the worst thing in scientific world is writing things others cannot read, and the authors did this by introducing many strange notations. These things makes the good work sometimes almost unreadable. This is not computer systems in which we use "cp" for the copy command and "cd" for change directory command.