Matthew Lane is a mathematician who maintains an interesting blog Mathematics Goes Pop! where he links mathematics to popular culture, and this book is perfectly in line with that. Six years before, Keith Devlin in his book Mathematics Education for a New Era: Video Games as a Medium for Learning (A K Peters, 2011), already argued that video games could be a tool for (mathematical) education. Devlin, as well as Lane have some sensible ideas about how to make use of the fact that gaming is so popular among youngsters into a tool or an incentive to learn some mathematics. That this can be obtained by especially designed or pimped versions of existing games is rather obvious, but Lane claims here that any game is suitable. Just analysing the winning strategies or the way in which adventurous problems have to be solved in different games can be concrete examples of a mathematical abstraction. Even, just the creativity that goes into exploring the possibilities within the rules of the game and the endurance with which it is played may promote an attitude of trial and error within the rules of mathematics and a culture of perseverance in solving mathematical problems. Anyway, it would be a waste if the popularity of gaming would not be exploited to serve a higher purpose.
In different chapters, Lane gives examples of how these ideas can be brought into practice by just relying on some popular video games that were not especially designed with an educational purpose in mind.
The first chapter introduces several games, in which physical reality is overly simplified. Gravitation and inertia are missing and worlds may be even just two-dimensional. However in the game A Slower Speed of Light, as you may have guessed, the speed of light is lowered so that one moves through the landscape and it will be observed just as relativity theory predicts when you are travelling close to the speed of light. In Miegakure the environment is the familiar three-dimensional setting, but one can move into a fourth dimension to avoid obstacles. Moving to a fourth space dimension is not possible in reality, but there is no problem to experience it in a video game. The gamer can experience a mathematical abstraction or a physical observation that is impossible in real life.
Chapter two is about guessing games like Family Feud where two teams have to guess the five most popular answers to some question. The popularity of these games dropped drastically after a short time because the number of questions was finite, and hence the questions keep repeating after a while. This can be the hook on which to attach some statistics and combinatorics and to design a procedure to avoid repetition as much as possible by attaching weights to questions that have already been asked. This is like interpolating between drawing balls from an urn with and without replacement, something that simply studying combinatorics mathematically does not offer.
The pitfalls of voting systems is another popular, yet tricky business to analyse mathematically. This applies not only to politics but also to games in which the user has to grade some components and also to the scores and the ranking of the users themselves. The way in which the player collects his points can be very complicated, and it may not always be clear what will be the score, positive or negative, that can be earned by their actions. Inverse engineering of your final score is not at all a simple problem. But if you succeed, then it should be possible to detect impossible scores, or perhaps screen configurations revealing partial information that is not possible, given the rules of the game. Of course the latter remotely refers to the consistency of a logical system. There are two chapters devoted to this kind of problems.
Chapter five is all about chasing and shooting. This is the chapter that is the most mathematical or at least the one with most formulas. As far as shooting is concerned, one may consider two kinds of missiles: those that go in a straight line and bounce off walls or the heat seeking missiles that lock in on the target and adapts its trajectory continuously. In the first case, the mathematics involves some simple trigonometry, but still the moving target complicates things, and it becomes really tricky when there are multiple reflections on walls. This is the part that has most of the formulas. The trajectories of heat seeking missiles are not piecewise linear anymore. They can in principle still hit a target that disappears behind a corner. This is a more involved issue and it is worked out to some extent in an addendum. But even a simple interception problem of an enemy missile moving on a straight line towards a target that has to be neutralised by your own missile, also moving in a straight line, is interesting to investigate. A blast with a certain radius can help you still destroying the enemy missile when the interception point is slightly missed. There are some quite interesting mathematics involved here.
As we progress in the book, the mathematics and the abstraction is cranked up a bit. The next chapter is about computational complexity and the P vs NP problem. These complexity concepts are introduced by explaining Kevin Beacon numbers. This is the distance of an actor to Kevin Beacon measured in coactor-of-coactorship. It is the analogue of the Erdős number which is the co-authorship distance from Paul Erdős, which is quite popular among mathematicians (I wonder why the Erdős number is not even mentioned). Finding these numbers is a shortest path problem in a graph and that is a problem from class P, but finding the longest path or the path of a certain length between two nodes are known to be NP-complete, i.e. easy to check but difficult to solve. So are some problems related to Tetris. Another well known example is the travelling salesman problem. This is a problem a gamer has to solve when he has to pick up some potions, treasures or weapons at fixed places in a maze. Finding a fast algorithm for solving them will earn you instant fame and a 1 million dollar prize from the Clay Mathematical Institute. Games in the class NP are usually the more challenging and perhaps therefore the more attractive ones. There is however little hope that you will crack the P vs. NP problem by playing video games.
There is a game called Sims which is all about getting (and keeping) friendship relations. Chapter 7 is about modelling such relations between two persons. Several models are proposed in discrete and in continuous time. The latter involves differential equations. It is not explained how to solve systems of differential equations, but solutions are plotted graphically, so that interpretations can be given. This moves seamlessly to the next chapter where nonlinear elements cause chaotic behaviour. For example when a third person competes with the second for the friendship of the first: a three-body problem. Chaotic trajectories may also result when a shell is fired that behaves like a ball on a billiard table. Even when these tables have simple geometries like squares or ovals or when there are a few obstacles inside.
In a final chapter Lane reflects on how video games can help in solving pedagogical issues. He explicitly refers to Devlin's book mentioned above and to other publications and reports on experiments that have been conducted at several places.
From this summary, it is clear that this is not about the mathematics of video games which would be much more involved with modelling the physics of the scenes, and the involved mathematics of computer graphics needed for rendering realistic characters. On the contrary, this is all about relatively simple mathematics and logical questions that the gamer could ask spontaneously or with a little help from his teacher. It's the mathematics hidden behind the game, the one not really explicitly visible. The game or its modes of operation can be the hook on which to hang the meaning of some abstraction or it can justify why a certain mathematical concept is useful. The mathematics itself is not really the focus of the book. Differential equations are mentioned but not their solution method for example. Lane just gives some examples of where a game can be an incentive to engage in a mathematical problem, and these problems go well beyond the cuddling mathematics of kindergarten. Lane is certainly convinced of the idea and he has a broad knowledge of the many different games, probably earned with a lot of experience. He does a good job in making his point and the ideas are not naive and they do make sense. If, as a teacher, you are game-phobic and feel like an alien in this virtual world of your students, don't be afraid of this book. Lane does a marvellous job in explaining what all these games do, or at least you are informed about what you need to know, and the book is amply illustrated. We shall not be teaching all our mathematics using games in the near future, but who knows what will happen when the ideas are elaborated further in games especially designed with an educational purpose. It is not unthinkable that they become standard ingredients in our educational toolboxes.