Math Problem Statement
Recall that a rook in chess attacks along rows and columns. Define R_n to be the number of ways of arranging n rooks on an n x n chess board so that no two can attack each other and their placement is symmetric about the diagonal from lower left to upper right. Which is a correct formula for R_n? R_n = n!/2 R_n = n(n-1)/2 R_n = R_{n-1} + (n-1)R_{n-2} R_n = nR_{n-1} + (n-1)*R_{n-2}
Solution
The problem involves arranging rooks on an chessboard such that their placement is symmetric about the main diagonal (the diagonal from the lower left to the upper right) and no two rooks attack each other. Let's analyze the possible formulas:
Key observations:
- Symmetry about the diagonal: If a rook is placed at position , another rook must be placed at to maintain symmetry.
- Non-attacking condition: No two rooks can be in the same row or column.
Understanding the structure:
- The positions of the rooks can be described as permutations of the set such that each number appears exactly once in the sequence, representing rows and columns.
- To ensure diagonal symmetry, the placement must follow the constraint that is paired with . This means a rook on forces a placement at .
Recursive analysis:
To compute , consider:
-
Base cases:
- : One rook fits on a board.
- : There is only one way to place two rooks symmetrically on a board.
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Recursion relation:
- When placing a rook at position , there are choices for .
- The remaining positions reduce the problem to placing rooks on an board.
- Additionally, off-diagonal placements may pair with and , introducing terms involving .
The correct recursive relation is:
This corresponds to the fourth formula:
Final Answer:
The correct formula is:
Would you like a step-by-step derivation or examples to illustrate this?
Related Questions:
- What are the explicit values of and based on the recurrence relation?
- How does the diagonal symmetry condition restrict the arrangement of rooks?
- Can be expressed in terms of factorials or combinatorial coefficients?
- What is the growth rate of as ?
- How does the problem change if we consider symmetry about other diagonals?
Tip:
To solve problems with recursive formulas, always analyze base cases and carefully account for all contributing terms in the recurrence relation.
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Symmetry in Chessboard Problems
Recursion
Formulas
R_n = n * R_{n-1} + (n-1) * R_{n-2}
Theorems
Permutation of Rows and Columns
Diagonal Symmetry Constraints
Suitable Grade Level
Grades 10-12 and above (Advanced Mathematics)
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