Guichard-Combinatorics-Graph-Theory · Problem 1.2.5
Fundamentals
Mathematics — Probability · combinations-permutations
Problem
In chess, a rook attacks any piece in the same row or column as the rook, provided no other piece is between them. In how many ways can eight indistinguishable rooks be placed on a chess board so that no two attack each other? What about eight indistinguishable rooks on a board?Intuitive explanation
The key insight is that if no two rooks attack each other, they can't share a row or column—so you need exactly one rook per row and one per column. This transforms the problem into counting permutations: once you decide which column gets the rook from row 1, which column gets the rook from row 2, and so on, you've completely determined a valid placement. For an board, you're really just counting ways to arrange objects, which is a factorial.
Intuitive explanation
The key insight is that if no two rooks attack each other, they can't share a row or column—so you need exactly one rook per row and one per column. This transforms the problem into counting permutations: once you decide which column gets the rook from row 1, which column gets the rook from row 2, and so on, you've completely determined a valid placement. For an board, you're really just counting ways to arrange objects, which is a factorial.