Question

Consider the following game between two players. The players take turns moving a rock among cells of an \(m \times n\) matrix. At the beginning of the game (before the first move), the rock is placed in the bottom-right cell of the matrix [cell \((m, n)\) ]. Player 1 goes first. At each turn, the player with the move must push the rock into one of the three cells above or to the left (or both) of the cell where the rock currently sits. That is, the player may move the rock to the cell above the current position, to the cell to the left of the current position, or to the cell diagonal to the current position in the up-left direction, as pictured below. A player may not move the rock outside of the matrix. The player who is forced to move the rock into the top-left cell of the matrix [cell \((1,1)\) ] loses the game. (a) Suppose the dimensions of the matrix are \(5 \times 7\). Does either player have a strategy that guarantees a victory? If so, which of the two players has such a strategy? (b) In general, under what conditions on \(m\) and \(n\) does player 1 have a winning strategy and under what conditions does player 2 have a winning strategy?

Short answer

For a 5x7 matrix, Player 1 has a winning strategy as 5+7=12 (even). Player 1 has a winning strategy when m+n is even; Player 2 has a winning strategy when m+n is odd.

Step-by-step solution

Understand the Game Mechanics

In this matrix game, two players alternately move a rock from its starting position in the bottom-right corner to either move one cell upwards, one cell to the left, or diagonally up-left. The goal is to avoid being the player who moves the rock to the top-left cell.(1,1), as this represents a loss.

Game Analysis for Specific Case (5x7 Matrix)

Start with understanding smaller grids to see patterns. For a 5x7 matrix (which has dimensions 5 rows by 7 columns), the game can be analyzed by considering each cell \(i,j\) as either a winning or losing position.

Strategy Determination for 5x7 Matrix

The essential concept is that a current player is in a losing position if all possible moves bring the opponent to a winning position. Conversely, if the player can move the rock to at least one losing position for the opponent, they are in a winning position. By following this logic, determine the pattern in this matrix.

Establish Patterns and General Strategy

For any given \(m imes n\) matrix, we can generalize the results by observing patterns in matrix games of smaller size and their outcomes. The strategy relies on whether \(m+n\) is even or odd.

Solution Based on the Outcome Patterns

From the analysis on smaller matrices leading to the 5x7 matrix or general dimension, it can be noted that if \m+n = even\, Player 1 has a winning strategy, and if \m+n = odd\, Player 2 has a winning strategy. For the specific 5x7 matrix, \(5+7=12\) (even), making it a winning strategy for Player 1.

Key concepts

Matrix Games

Matrix games are situations where two players compete to reach a goal or avoid a loss by strategically moving a piece on a grid-like structure, such as an \(m \times n\) matrix.

The goal in this specific matrix game is not to be the player who is forced to move the rock into the top-left cell \((1,1)\). This setup is known as a zero-sum game because one player's gain is fully offset by the other's loss.

Some key characteristics of matrix games include:

• Grid Structure: The playing field consists of rows and columns forming a matrix.
• Turn-based Play: Players alternate in making moves, requiring strategic planning.
• Constrained Movements: Methods of moving are limited to specific directions, which requires careful forethought by each player.
• Outcome-Dependent Moves: The end game can be reached with a specific series of moves, allowing for the emergence of patterns and strategies.

This game exemplifies fundamental game theory concepts, where players aim to reach winning positions through strategic advances, careful planning, and anticipation of opponents’ moves.

Winning Strategy

In game theory, a winning strategy allows a player to force a victory regardless of the opponent's moves. To win, a player must place their opponent in a consistently advantageous position for themselves. In this matrix game, both players strategize to avoid being the one forced into the losing move.

For Player 1 in an \(m \times n\) matrix, a winning strategy is defined by the sum \(m+n\).

• If \(m+n\) is even, Player 1 can force the opponent into losing positions consistently, guaranteeing a win.
• If \(m+n\) is odd, it's Player 2 who can consistently turn situations to their advantage, leading Player 1 into a losing position.

The ability to discern whether \(m+n\) is odd or even allows determination of which player holds the advantage in the game, thus creating a plan to maintain that position and avoid disadvantageous moves.

Game Analysis

Analyzing this matrix game involves breaking down the movement patterns and studying smaller grid structures for understanding. This game analysis helps establish which cells are winning or losing states. The approach involves:

• Identifying Initial Conditions: Start from the smaller sections of the matrix to see which configurations lead to obvious win or loss outcomes.
• Establishing Patterns: Look for recurring conditions whereby moving to specific cells consistently results in a win or a cascade of beneficial moves.
• Testing Moves: By considering all potential future moves, one player can try to force the opponent into losing moves.
• Predicting Outcomes: Using previous observations, players can predict which states to aim for and which to avoid, based on the sum \(m+n\).

Through this analytical lens, players determine their overall approach to achieve and maintain advantage until the win or to avoid setting themselves up for loss.

Optimal Strategy

An optimal strategy in this context involves leveraging every movement possibility to minimize the opponent's advantages while maximizing one's own.

Achieving the optimal strategy requires:

• Understanding Matrix Patterns: Knowing the nature of winning and losing cells and being prepared to exploit the board's patterns.
• Preemptive Positioning: Moving the rock in a manner that anticipates potential outcomes and restricting opponent options.
• Strategic Flexibility: Being adaptable to force or evade landing in critical cells, thus modifying the course to ensure the opponent fails to secure a winning path.

For instance, using the manageable insights that if \(m+n\) is even, Player 1 seizes the advantage suggests Player 1 can premeditatedly control the game flow.

The use of such a strategy ensures maintaining superior positioning, controlling the matrix's traversal path, and ideally forcing the opponent into the unwelcome position of having no winning move available, leading to their eventual loss.