A 6-by-6 chessboard is perfectly covered with 18 dominoes....

Name: Problem 25, Chapter 1: Introductory Combinatorics
Uploaded: 2019-11-26T23:25:51-08:00
Duration: 1 min 33 s
Channel: James Chok
Description: A 6-by-6 chessboard is perfectly covered with 18 dominoes. Prove that it is possible to cut it either horizontally or vertically into two nonempty pieces without cutting through a domino; that is, prove that there must be a fault line.

A 6-by-6 chessboard is perfectly covered with 18 dominoes. Prove that it is possible to cut it either horizontally or vertically into two nonempty pieces without cutting through a domino; that is, prove that there must be a fault line.

Key Concepts

Domino Tiling

Domino tiling involves covering a grid completely with 1×2 or 2×1 rectangles (dominoes) such that each domino covers two adjacent grid cells and the entire region is covered without overlaps or gaps. This concept is central to many combinatorial and geometric problems where spatial arrangement and covering arguments are crucial.

Fault Line in Tiling

A fault line in a tiling is a line (either horizontal or vertical) that can divide the tiled board into two nonempty parts without cutting through any tile. This concept is important because identifying such a line demonstrates a specific structural property in the tiling arrangement that can be used to draw conclusions about the overall configuration and its flexibility.

Checkerboard Coloring and Parity

Checkerboard coloring and parity arguments involve alternately coloring the grid cells and then using the parity (balance) between colors to reason about the placement of dominoes. Since each domino in a standard tiling covers one cell of each color, these methods provide powerful invariants that help in proving properties about tilings, such as the existence of fault lines.

Invariants in Combinatorial Configurations

Invariants are properties that remain unchanged under certain transformations or operations. In domino tilings, invariants such as the number of dominoes covering pairs of differently colored cells are used to ensure that any complete tiling must satisfy certain conditions. This approach can also help in identifying necessary cuts or fault lines without disrupting the overall tiling.

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