Introduction

Introdution

Games have existed in human society for a long time, at the very least since the neolithic era four to eight thousand years ago. The word “games” has a quite broad interpretation. All games have a ludic aspect and all are an imaginary combat between or among a group of adversaries. Athletic games like soccer have been played in distinct societies like ancient Greece and China, as well as in primitive Latin America. Other kinds of game are theatral games and musical games, also quite old. Finally, we have games with specific rules which involve rational thinking, like chess and Go.

Games like these are essentially different from sportive or artistic games. Even physically impaired people may be well prepared to play such games – they demand mental and not physical capacity. Games like these are denominated mathematical games.

Books about various different mathematical games have existed for centuries, but rarely about the principles behind mathematical games themselves. In 2007 a group of teachers and students at the Federal University of Goiás was created that began both the abstract study of mathematical games and the study of their use as a pedagogical tool at the elementary and high school level.

It's common to not understand the diference between the theory of mathematical games and the classical study game theory. This second area began in the first half of the twentieth century with the work of mathematicians like John Nash, Morgenstern and von Neumann. This theory considers games between two or more participants who search for strategies to maximize their profit or benefits in interactions with the others. To win these games implies gaining as many or more points than other players, generally in an economic or military sense. Normally, one player will not have complete information about his or her “adversaries”. Obviously this is very different from a game like chess or Go, where the information is complete and players play in sucession one of the other, and in which each player desires to simply “close” the game, denying his opponent any move at all.

The Concept of a Mathematical Game

A great number of games will be discussed in this site. In general all will have the following aspects in common:

1.they always involve two players
2.there is no tie possible
3.to win implies being the last to play

It's true that in chess and some other games, ties are possible, but it's an aspect that we prefer to ignore for now to have a more rigorous analysis of game play. Also, the aspect (3) is only one possibility, existing another: to win could mean to not be the last to play. Games like this are called games with the option misère, and will be considered in due time.
Notice that in these games

1.there is no role for luck in the game – there is total transperancy
2.the games involve only perception, reason and logic.

Besides the above there is one more possible distinction between games. The games that have distinct “armies”, like chess and checkers, are called partesan. Games that do not have distinct teams, like in Nim and Spouts, are impartial.

The Mathematics of Games

Consider a match between two players, A and B. In a “partesan” game two different armies will fight a battle throughout the game with alternating moves. Each move should be the implementation of a certain strategy – with or without error in planning – and simultaneously a response to the adversary's moves. In a certain moment the game will evolve into a situation in which one of the players, say A, has no possible response to player B. A would be in a situation of disadvantage: independent to what he or she does - if B commits no mistake, B will win. Obviously it could also be the opposite situation: whatever B does it is to no avail, A can certainly win.

We call a game with advantage to A positive and one with advantage to B negative. The games we will discuss called Hex and Domineering are partesan, as is Hackenbush (to be briefly discussed as well). In these moments of the game the value of the configuration will be positive or negative.

It should be clear that at the beginning of the match there is no intrinsec advantage on anyone's behalf. In mathematical games this occurs in a well-defined way, mathematically speaking. In other words, no “luck” is involved. In the 1930's a theorem was proven by two European mathematicians independently, Sprague and Grundy. The theorem of Sprague-Grundy says that in each moment along a match of a mathematical game each player will have an optimal strategy to follow. If both players follow their strategy the situation is always one of the following:

By fuzzy we mean a value that, by what was said above, is neither positive nor negative nor zero – in short fuzzy corresponds to a different kind of value. It means that whoever plays first has an advantage. In impartial games, with no distinct armies in action, the value of a configuration will always be either zero or fuzzy, because there is no real difference between the players. It will always be a question of who plays next. These fuzzy values, mathematically, correspond to what are called surreal numbers.

Partesan games are more complicated. In them the value may be fuzzy or zero but may also be positive or negative.

To view a detailed fair game, go to - Nim , Kayles ou Sprouts. To see a game partesan go to-Hackenbush
or Hackenbush for Kids.