Natasha Glydon

People who enjoy working through puzzles, like crosswords or word searches, are enhancing their mathematical minds as they play.  Games of strategy and logic have mathematical foundations and encourage players to think about various mathematical concepts including patterns, abstraction, sequencing, and combinations.  Often, people play these types of games and do not even realize they are developing and refining their math skills.

Chess

Chess is a board game that requires its two players to make moves against each other in hopes of capturing their opponent’s king.  It is a game of strategy that develops memory and concentration.  In order to play chess successfully, players need to “attack” their opponent’s pieces and eliminate them.  Chess involves a lot of problem solving.  Your mind is constantly scanning the board and piecing together possible solutions.  Chess challenges you to explore new combinations and anticipate the result.

Chess is about thinking logically.  There are six different types of pieces, each moving in a different direction, over a different amount of spaces.  A chessboard is symmetric with a diagonal line of symmetry.  The chessboard consists of eight rows by eight columns, alternating in color, and can be seen as a plane.  Players learn and master how the pieces move through the plane and can apply what they know about mathematics to make beneficial combinations of moves.

 Image reproduced with permission of 3D Chess

The chess pieces create different line patterns on the board since each piece moves differently through the plane.  A bishop, for example, moves diagonally across the board in any direction.  A bishop starting on a white space and a bishop starting on a black space will never meet.  This is similar to the concept of parallel lines in a plane.  Knights move in an ‘L’ shape; one space left or right and three spaces up or down, and vice versa.  When knights move, they end up on a square that is in a different row, a different column, and is on a different colored space.  In this way, players use their problem solving skills to make successful combinations of moves.

While playing chess, it is possible to determine who is winning mid-game by assigning numerical values to pieces still in play.  A king is worth 0 points, a pawn is worth 1 point, a knight is worth 3 points, a bishop is worth 3 points, a rook is worth 5 points, and a queen is worth 9 points.  These numerical values are determined by the versatility of the pieces.  The queen is the most valuable piece on the board since she can move in any direction, any number of spaces.  This is why the queen is worth the most points.  Not only can this determine who is winning, but it can also be a valuable counting hint for players.  For example, a player may sacrifice two pawns in order to capture a bishop.  Ultimately, the player will be ahead one point, suggesting that is was a productive move, although there is no real point system when playing chess.  In much the same way, players can look for alternatives instead of sacrificing too much.  Players can use math to help them problem solve.

Sudoku

Sudoku is a logic puzzle.  It requires thought and concentration, similar to chess.  It is an independent game, played on a square board, usually 9 x 9.  The idea is to arrange the numbers from 1 to 9 in each row, column, and 3 x 3 region.  This is to be done mathematically rather than using a guess and check method.   When solving Sudoku, one must use logical analysis and a process of elimination to deduce what values are to be placed in which spot.  Each Sudoku has a unique solution.  Sudoku is about making guesses and seeing their result.  When working through Sudoku, the numbers can be thought of as permutations, since we are looking at arrangements of numbers.  Sudoku enhances skills of pattern recognition, abstraction, and problem solving.

Combinatorial Game Theory

Most two-person games that require movements of pieces can be considered mathematical.  Tic-Tac-Toe and Mancala are examples.  There are strategies involved.  Players need to think about problem solving, patterns, and counting, all of which are mathematical concepts.  Each move determines which moves will be possible for the player’s next turn.  Players must use their problem solving skills to think ahead and predict what moves their opponent will play.  The analysis of these game strategies is called Combinatorial Game Theory (CGT), which is essentially studying different winning combinations and strategies.   CTG looks at the optimal sequence of moves for each player in order to win the game.

 Image reproduced with permission of Eddie Matejowsky

Strategic games make the mind work logically and creatively.  When working through these games, the mind and brain are practicing problem solving and logic skills, which are important in developing proficient math skills.  Puzzles and problem solving are very common mathematical concepts and can be seen in other real world applications, aside from just games.