Preface

Games both as activities and as a basic educational tool are not only important from birth to death, but have been so from the beginnings of human society to the present day. The discipline of game theory deals mainly with types, description, algorithmization and strategies, but also the formalization of games. Indeed, games and the gamification of some human activities on various platforms have become an important part of our daily lives. Even philosopher and pedagogue J.A. Komensky spoke of "a school through play". Games in all shapes and forms contribute to the development of society. Today's young people will have played at least one game during their life, and as technologies develop, more people are playing games than ever before.

Modern game theory as a science is a very broad area. This book seeks to reveal part of what lies behind the "curtain" of game theory as a whole. It describes some game types, game theory tasks, the math background to game theory, game semantics, rule development, game goals or objectives, and the mechanics, design and development of the field, including computer games. The book includes, but is not limited to, a description of multiplayer and single-player games, cooperative and non-cooperative games, symmetric and asymmetric games, simultaneous games, turn-based games, complete and incomplete information games, scoring systems, and optimal play or winning strategies. Further topics include optimization of turns, time games and game complexity, games testing, games evaluation, games presentation, and the social, economic and business impact of games. Finally, there are the education and training aspects of gamification, or learning through playing.

I would like to express my sincere gratitude to all the authors and co-authors for their contributions. The successful completion of this book has been the result of cooperation between many individuals. I would especially like to thank IntechOpen Publishing Process Managers Marina Dušević and Dragan A. Miljak for their support.

> **Branislav Sobota** Technical University of Košice, Košice, Slovakia

**1**

**Chapter 1**

Games

*David Ginat*

**Abstract**

Problem Solving of Mathematical

Mathematical games are problems that involve algorithmic solutions. The solutions require recognition of hidden patterns and capitalization on these patterns. The natural tendency of many problem solvers is to devise algorithms without fully unfolding patterns. Such an approach lacks rigor and may lead to undesired outcomes. This chapter underlines a rigorous approach, of first focusing on the characteristics of a posed game and then developing its algorithmic solution. The solution development "goes" hand-in-hand with the realization of correctness. The approach is based on declarative observations, which capture the "what" of patterns prior to the "how" of gamestrategy instructions. We illustrate the approach with colorful mathematical games of different characteristics and underline elements of solution processes, including

creativity, problem-solving features, and mathematical notions.

operational perspectives, invariance

core of mathematical problem solving [2].

**1. Introduction**

**Keywords:** problem solving, Mathematical games, creativity, declarative and

Mathematical games are mostly two-player games. Their specifications involve initial positions (states) and sets of rules for the players' moves. The two players play in alternating turns, and the game ends upon reaching a final position. The winner is usually the player who makes the last move. Most games are *complete information games*, in the sense that the game information is known at any given time to both players. The assumption is that each player always makes the best move available for her. Positions that lead to a win are *winning positions* for the player who moves next, and positions from which one will lose, regardless of her following moves, are *losing positions*. The player who moves first is *the first player*, and the other is *the second player*. In most games, an *a priori* analysis of the game may "tell" which initial positions are winning positions and who will be the winner. The challenge for problem solvers is to employ an analysis that will reveal for each initial position whether it is a winning or a losing position and to offer a sound playing strategy for winning the game [1]. The winning strategy involves algorithmic instructions, based on unfolded underlying patterns. Recognition of hidden patterns and capitalization on patterns are the

The primary inclination of many problem solvers is to invoke an *operational perspective* [3, 4], in which they focus on the instructions for the winner to follow (e.g., "if the
