Strategy Games to Enhance Problem-Solving Ability in Mathematics

Alfred S Posamentier, Stephen Krulik

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789813146365
• Topic: Matematica
• Subtopic: Giochi in matematica

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Games are seen only for recreation. However, this book shows that games can be used to strengthen problem-solving skills and beyond. This book presents strategy games and discusses for each one solutions towards a winning position in the game. In most cases, these strategies are analogous to problem-solving strategies in mathematics. Readers are also exposed to a wide variety of games from several different cultures, which will broaden the perspective of the readers.

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Contents: An Introduction to Strategy Games;Tic-Tac-Toe Games;Blocking Games;Games Requiring an Ongoing Change of Strategies;Miscellaneous Strategy Games;Suggested Solutions for Selected Games;Game Boards for Various Strategy Games;
Readership: Students and general public interested in mathematics in strategy games and problem solving.
Mathematics, Problem-Solving, Techniques, Strategies, Games

• The books presents a wide range of games, some of the strategy games are not known to general public
• Aside from presenting a starting strategy, the book then presents the best winning strategy for each game
• Unique approach to considering problem-solving methods camouflaged in game-winning strategies

Chapter 1

An Introduction to Strategy Games

One of the most important skills in today’s society is problem solving in mathematics and beyond. The ability to resolve problem situations is paramount in today’s world. There has been a great deal of research done, which reveals a strong connection between problem solving and playing strategy games. We know that people who are good at problem solving are usually good strategy-game players. Conversely, people who are good at playing strategy games usually show an intuitive approach to successful problem solving. Therefore, throughout this book, we will be presenting a wide variety of strategy games, each of which mirrors in some fashion the kind of thinking that a mathematician engages in during the process of problem solving. It is through this kind of thinking that one enhances the thought process which allows for a strengthened problem-solving capability. Though quite entertaining strategy games are also a training ground for problem solving. Onward now we introduce the process and the various exemplary models of strategy games.

1.1. What is a Strategy Game?

Most people today are used to playing one or more electronic games on devices such as the computer, the i-pad, the smart phone and so on. While these games are indeed enjoyable and do require a certain amount of skill and eye-hand dexterity, many do not require — nor even encourage — careful thinking and strategy development. In some games, time is a major factor. The player is competing with an unknown opponent or against the computer. The two-person strategy games are different. Each game is a head-on, direct competition between two people, each trying to win a game that requires thought and development of a winning strategy. It is true that a person can win some of the time by random play or luck, rather than skill. But this will not happen consistently. After all, it takes time to develop a winning strategy, and when the two players are equally skilled (or unskilled, as the case may be) either player can sometimes win. However, in a two-person strategy game, each player is encouraged to utilize a deeper concentration on developing a needed strategy to succeed at winning the game. Developing a winning strategy should lead to continued success in that game. Finding a strategy that enables one to solve a problem often leads to a general strategy for solving similar problems. In a like manner, as in problem solving, developing and applying a successful strategy at one game may help win at several similar strategy games. Of course, when both players discover the same winning strategy and apply it correctly, many of these games often end in a tie.

1.2. Why Strategy Games?

First of all, we should ask why games? People enjoy playing games. Games are fun! Games are challenging! They provide an opportunity for developing skills and techniques to be successful. People are used to playing games. They expect there to be rules of the game, and a goal for the game. The connection between strategy gaming and mathematical problem solving becomes readily apparent when we examine the heuristics of each, side-by-side. When playing a strategy game, each player must ask oneself some very basic questions, much as is done when engaging in problem solving. Following, we have placed each strategy-gaming heuristic skill directly alongside the corresponding problem-solving heuristic skill to allow for a quick comparison.

Strategy Gaming	Problem Solving
1. What are the rules of the game?	1. Read the problem
• What constitutes a “win” or a “loss”? • What is meant by a “move”?	• What’s the question? • What are the facts?
2. What is a good opening move?	2. Select an appropriate strategy
• Is there an advantage in going first? • Should you play defensively?	• Apply the strategy
3. What strategy can lead to a win?	3. Solve the problem
4. Does your strategy work all the time?	4. Reflect or Look back
• Do you win consistently?	• Does the strategy work on similar problems?

Notice how the strategy-gaming heuristics are similar to those of problem solving. In fact, the heuristics are used over and over throughout the entire game. Each time a player makes a move in a strategy game, that player presents the opponent with a new problem. Players constantly ask themselves, “How do I counter that move with one of my own?” This is all part of developing a game strategy.

1.3. Finding and Creating Strategy Games

There are many books that contain one- or two-person strategy games. Some of these games will be discussed in this book. However, you can often create some of your own games using either or both of the following techniques:

•Change the goal of the game.

•Change the shape of the given playing surface, game pieces, or game board.

Figure 1.1. Tic-tac-toe board.

Let’s take an example of each. Most people are familiar with the basic game of tic-tac-toe (sometimes called “Noughts and Crosses” in Great Britain). About 500 BCE, Confucius describes a game called Yih, which is today’s tic-tac-toe. Yet in the Western world the game seems to have its origins in ancient Rome and Egypt and has come to its current form in the nineteenth century. In this game, players alternate turns placing an “X” or an “O” in an empty cell on a 3 × 3 or 9-cell grid (see Figure 1.1).

The goal is to get three of your own marks in a straight row, with no intervening spaces or opponent’s mark. Most of us have played this game and developed a strategy that will lead to either a win or a draw. The majority of players usually prefer to go first. Since there are nine cells on the board, going first ensures a chance to place your mark in five cells to the opponent’s four cells. Furthermore, most players going first place their mark on the center square. This square is involved in a winning position 4 times out of 8. Corner squares are involved in a winning position 3 times out of 8. Sounds easy enough, doesn’t it? And yet, the center cell as a first move is not the best approach. The player going first should place their X or O in one of the four corner cells. This corner-cell placement is the one that can lead to developing two possibilities for a win. Once these two possibilities have been developed, the opponent cannot block both at once, and a win is assured. If both players are careful and know the strategy, then the very least that can occur is a tie.

Let’s inspect the strategy used in playing tic-tac-toe. The player to go first, using the X, has three options for his first move. He can either place the X in a corner cell, a middle border cell, or in the center cell. This player can force a draw or can win with any of these three starting moves. However, by placing the first move in a corner cell, the second player is limited in his move to avoid losing.

The second player, using an O, must respond to the first player’s move defensively. If the first player chose to place the X in a corner cell, then the second player must place the O in the center cell. If the first player chose to place the X as the initial move in the center cell, then the second player must place his O in a corner cell. If the first player places his initial move by placing the X in a side middle cell, then the second player must place the O either at the center cell, or at a corner cell next to the O-placed cell, or at the side middle cell opposite the X cell. Any other move by the second player will result in a win by the X player. After these two moves, the second player continuously places Os to block the first player’s attempt to get three Xs in a row. Of course, there is always the possibility that the first player, X, might make a careless move allowing the second player, O, to win rather than just to force a draw.

For the second player, placing O, to guarantee a draw, he would need to adhere to the following: If the first player does not place the X in the center cell, then the second player should occupy the center cell with the O, followed by a side middle cell. If the first player places an X in the corner square, and the second player places and O the opposite corner, that will allow the first player to win if he places the second X in one of the occupied corners. If the first player places an X in the corner cell, then the only way the second player in force a tie to place an O in the center cell and then, as his next move he would place an O in a side middle cell. By this time you should be able to notice the various strategies that clever players would use to always end up in a tie. It is when one player makes a careless move the other player will be able to win.

However, let’s make a simple change in the game, thus creating another game and requiring another strategy. In this new version of the game, three-in-a-row loses the game. Does the strategy change? Do you still want to go first? Do you still want to place your X or O in the corner square? A new problem has been presented, and a different strategy must be developed. Play the game with this new goal and see what transpires.

A second approach to creating a different game is to make a change to the game board. In a new game of tic-tac-toe, we might play on a 5 × 5-cell square board, but require 4 in a row to win. Now does the strategy change since there are 25 cells? What’s the strategy here? Or, perhaps, change the board to a 4 × 4-cell board of 16 cells, and try to get three-in-a-row. Now does it make sense to go first? After all, each player will have a chance to fill eight cells. Should we still play 3 in a row wins the game? Or 4 in a row? The strategy for winning in each version of the game has changed! We will consider these variations later in the book.

Even a strategy game as simple as original tic-tac-toe affords many opportunities to utilize mathematical skills. Explaining the rationale for making a particular move helps develop reasoning skills. The communicating of these ideas verbally, creates an informal dialogue, which leads to increasing the higher order thinking and problem-solving skills. Every move requires an informal calculation of the probability of winning or losing the game. Even an opponent’s move raises the question of the probability of winning the game.

This book is divided into chapters, each considering a different type of strategy game. Sometimes a single strategy can resolve several games, similar to applying a specific problem-solving strategy to several different problems. Other times each game may require a specific strategy. In some cases, we might provide a hint or a suggestion. We will not, however, reveal a complete strategy for a particular game until the last chapter of the book (Chapter 6), so that you can have time to craft your strategy and then if you wish, compare it to our strategy in the last chapter. We don’t want take away the fun of you, the reader, trying to develop your own strategy. Where a specific type of board is given, feel free to copy the board, enlarge it, and play the game as many times as it takes to discover a winning strategy.

Many of the games in this book have their origins in countries around the globe. Some have been played for centuries by children and adults alike. The players use game boards drawn on the sand or dirt, or painted on leather animal hides. Rocks or pebbles of different shapes or colors make up the playing pieces. Some games are available commercially with more permanent game boards made from wood, plastic or metal.

Most of the games in this book do not require any special equipment. Generally, all you need is a pencil, paper, a copy of the particular game board (most game boards are available in the Appendix following Chapter 6), and possibly some toothpicks, chips or coins, which should be easily identifiable for each player. We wish you lots of enjoyment.

Chapter 2

Tic-Tac-Toe Games

In this chapter, we will consider some variations of ...

Table of contents

• Cover
• Halftitle
• Series Editors
• Title
• Copyright
• Dedication
• Contents
• About the Authors
• Chapter 1 An Introduction to Strategy Games
• Chapter 2 Tic-Tac-Toe Games
• Chapter 3 Blocking Games
• Chapter 4 Games Requiring an Ongoing Change of Strategies
• Chapter 5 Miscellaneous Strategy Games
• Chapter 6 Suggestions and Strategies for Selected Games
• Appendix Game Boards for Various Strategy Games
• Index