Question

How many ways can 12 black pawns be placed on the black squares of an \(8 \times 8\) chess board? How many ways can 12 black pawns and 12 white pawns be placed on the black squares of an \(8 \times 8\) chess board? Half the 64 squares are black and half are red. No black (red) square shares an edge with a black (red) square.

Short answer

\( \binom{32}{12} \) for black pawns only; \( \binom{32}{12} \times \binom{20}{12} \) for both black and white pawns.

Step-by-step solution

Count Black Squares on the Board

An \(8 \times 8\) chessboard is divided into two colors: black and red. The black and red squares are arranged in such a way that no two squares of the same color share a side. Thus, there are precisely 32 black squares on the board.

Calculate Arrangements for Black Pawns

To find the number of ways to place 12 black pawns on the 32 black squares, we need to choose 12 squares out of 32. This can be determined by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of black squares (32) and \( r \) is the number of pawns (12). Calculate \( \binom{32}{12} \).

Calculate Arrangements for White Pawns Given Black Pawns

Once the black pawns are placed, 32 black squares minus 12 occupied squares leaves 20 available black squares. Next, choose 12 out of these 20 squares for the white pawns. This is given by \( \binom{20}{12} \).

Combine Arrangements for Both Black and White Pawns

To find the total number of ways to arrange both sets of pawns, multiply the number of ways to arrange the black pawns by the number of ways to arrange the white pawns: \( \binom{32}{12} \times \binom{20}{12} \).

Key concepts

Combination Formula

The combination formula is a fundamental concept in combinatorial analysis, used to determine the number of ways to select a group of items from a larger set, where the order of selection does not matter. This is particularly useful when you're dealing with problems that involve picking a subset from a collection. The formula is represented in mathematical notation as \( \binom{n}{r} \), where:

• \( n \) is the total number of items in the set.
• \( r \) is the number of items you wish to choose.

The combination formula is calculated using the following expression:\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]Here, \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \). The factorial function \( r! \) and \( (n-r)! \) are computed similarly.
In the context of the given problem, the formula helps us find the number of ways to choose 12 black pawn placements from the 32 available black squares on the chessboard by calculating \( \binom{32}{12} \). This indicates selecting 12 specific squares for the pawns, with all arrangements being valid as long as the number and type of squares align.

Chessboard Problems

Chessboard problems are intriguing puzzles that often involve arranging or calculating patterns based on the unique layout of a chessboard. An \( 8 \times 8 \) chessboard like the one in this exercise has 64 squares, half of which are black and the other half red. The board is designed so that no two adjacent squares are the same color.
For the task of placing pawns, only the black squares are considered, making it a classic example of highlighting and using specific parts of a structure. The problem of arranging pawns can be understood as managing these plentiful combinations on the pre-determined checkerboard layout. Each black square essentially serves as a possible location for a pawn, allowing us to explore different strategies and counting possibilities while using both combinatorial analysis and logical problem solving.

Discrete Mathematics

Discrete mathematics is a broadly applicable field that studies mathematical structures that are fundamentally discrete rather than continuous. This includes topics such as logic, set theory, graph theory, and combinatorics. Discrete mathematics is the backbone of these chessboard and combinatorial problems as it involves structures like the black and red squares, and pawn positions that have distinct and separate entities.

• Each black square on the board can be thought of as an individual set element.
• Pawn arrangements on the board involve counting and combinations, typical of combinatorial techniques.
• Solving these problems necessitates understanding and applying discrete concepts like systematic counting and permutation without the need for calculus-based continuity.

Understanding the foundational discrete structure of the chessboard allows us to efficiently handle and calculate potential strategies, making it an essential part of the thought process behind solving these kinds of exercises.