Consider a board with forbidden positions which has the...

Name: Problem 12, Chapter 9: Introductory Combinatorics
Uploaded: 2025-08-03T17:13:07-08:00
Duration: 2 min 31 s
Channel: Nick Johnson
Description: Consider a board with forbidden positions which has the property that, if a square is forbidden, so is every square to its right in its row and every square below it in its column. Prove that the chessboard has a tiling by dominoes it and only if the number of allowable white squares equals the number of allowable black squares.

Consider a board with forbidden positions which has the property that, if a square is forbidden, so is every square to its right in its row and every square below it in its column. Prove that the chessboard has a tiling by dominoes it and only if the number of allowable white squares equals the number of allowable black squares.

Key Concepts

Domino Tiling

Domino tiling refers to the process of covering a board completely with 1×2 or 2×1 domino pieces such that no domino overlaps another and there are no gaps. This concept is closely linked with combinatorial tiling problems and involves conditions on the board's structure and geometry to admit such a tiling.

Chessboard Coloring and Parity

Chessboard coloring is a method of assigning two colors (typically black and white) in alternation to the squares of a board, leading to a bipartite structure. In problems involving domino tilings, each domino covers one square of each color. Thus, a necessary condition for a tiling is that the number of allowable squares of each color is equal, establishing a parity condition that must be satisfied.

Perfect Matching in Bipartite Graphs

A perfect matching is a set of edges in a graph that covers every vertex exactly once, and in the context of domino tilings, there is a one-to-one correspondence between tilings of the board and perfect matchings in the associated bipartite graph formed by the chessboard coloring. This graph-theoretic interpretation helps in establishing necessary and sufficient conditions for the existence of a tiling.

Forbidden Positions and Monotonicity Constraint

The board described involves forbidden positions that have the property of monotonicity—if a square is forbidden, so are all squares to its right and below in that row and column. This constraint simplifies the configuration of allowed positions and plays a crucial role in analyzing the board's structure, enabling arguments based on induction or invariants to show the equivalence between the tiling existence and the balance in the number of allowed squares of different colors.

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