Question

A king is placed on the bottom left-hand square of an \(8 \times 8\) chess board and is to move to the top right-hand comer square. If the piece can move only up or to the right, how many possible paths does it have?

Short answer

The king has 3432 possible paths.

Step-by-step solution

Understanding the Problem

We have an 8x8 chessboard where the king starts at the bottom-left corner, which is position (1,1), and needs to reach the top-right corner at position (8,8). The king can move only up or to the right. We want to find out how many different paths the king can take.

Setting Up the Matrices

We will set up an 8x8 grid to represent the chessboard. We need to fill in this matrix with numbers representing the number of ways to reach each square from the starting point using only right and up moves.

Initialize the First Row and First Column

Since the only way to reach any square in the first row is to move all the way to the right from the starting position, each square in the first row has exactly 1 way to be reached. Similarly, each square in the first column has 1 way to be reached by only moving up. Therefore, fill the first row and first column of the matrix with 1s.

Combining Moves

For each square not in the first row or column, calculate the number of paths by adding the number from the square directly above and the number from the square directly to the left. The sum of these two numbers gives the total number of paths to reach that square. This is based on the principle that the king can arrive at a particular square either from the square immediately to its left or from the square immediately below it.

Filling the Matrix

Proceed to fill the matrix row by row. For example, starting from (2,2), fill each square using the sum of the number above it and the number to the left of it, until the last square at (8,8) is reached.

Finding the Answer

Once the matrix is completely filled, the number at the position (8,8) represents the total number of possible paths from (1,1) to (8,8).

Key concepts

Chessboard Problems

Chessboard problems are a classic area in combinatorics. They involve finding ways to solve various mathematical puzzles on a chessboard. Typically, an 8x8 board, the same as used in standard chess games, sets the stage for these problems. Understanding the specific moves of chess pieces helps in setting up initial conditions and solving problems effectively. For example, in the classic question about moving restrictedly across a chessboard (like the problem with the king), it’s crucial to know the boundaries and permissible moves.

• The problem often simplifies to a grid or a matrix operation, deciding how many ways a piece can move.
• Chessboard puzzles, like the one with our king, can involve creating pathways or avoiding certain squares.
• The mathematical representation of the chessboard allows us to use tools from combinatorics to solve these puzzles systematically.

Chessboard problems cultivate critical thinking and spatial awareness, often requiring an overlap of math, strategy, and prediction.

Matrix Paths

Matrix paths are a method to solve many movement problems on grids. Here, the grid or matrix doesn't just represent the board; it’s filled with numbers. These numbers define how many ways you can reach each square. It’s an excellent visual technique to understand path counting in combinatorics.
For our problem, the matrix represents the chessboard, where each cell (square) corresponds to how many possible journeys from the start can be made to that square.

• Each move either leads you up or to the right, incrementally increasing your path count.
• The top-left number in any square gives the cumulative paths leading there from the beginning.
• The approach is systematic, simplifying complex movement problems into basic arithmetic operations.

This method is useful not only in chessboard problems but in any situation requiring a clear calculation of multiple paths.

Path Counting

Path counting is a foundational technique in combinatorics. It helps determine the total number of paths possible between a start and an endpoint, given specific movement restrictions. This is particularly useful in problems like our chessboard king problem.
By counting paths, we create a strategy rooted in simple addition—that's checking movements originating from adjacent cells (either above or to the left).

• Initial paths start as unique (a single straight line up or across)
• Every node or point in the matrix accumulates the sum of potential paths from its accessible routes.
• Ultimately, you'll derive a total path count by the time you get to your endpoint

Combinatorics principles, particularly binomial coefficients, often play into these calculations when solving path-counting problems step-by-step.

King's Moves

A king in chess moves differently compared to other pieces. On the board, the king can move one square in any direction. However, in our problem, to simplify path counting, the king's movement is restricted to only move up or to the right.
By imposing these restrictions, we're channelling the movement to a combinatorics problem, focusing solely on movement roles that fill an 8x8 matrix.

• The emphasis on up and right movement mirrors approaches in other grid-based problems, steering clear of more complex moves.
• Counting paths requires considering only two directional options while reaching each new square.
• This simplifies calculation, turning a complex board journey into a manageable arithmetic problem.

In «normal» chess, moves are strategic and varied; yet, this clear directionality on movement highlights how structured and solvable grid navigation problems can be with restrictions.