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Omni-MATH / Let n>1 be a positive integer. Each unit square in an n×n grid of squares is colored either black or white, such that the following conditions hold: - Any two black squares can be …

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problem

Let n>1n>1 be a positive integer. Each unit square in an n×nn \times n grid of squares is colored either black or white, such that the following conditions hold: - Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge; - Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge; - Any 2×22 \times 2 subgrid contains at least one square of each color. Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of nn).
Plain-text mathematical notation (without MathML)
Let n>1 be a positive integer. Each unit square in an n×n grid of squares is colored either black or white, such that the following conditions hold: - Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge; - Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge; - Any 2×2 subgrid contains at least one square of each color. Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of n).
Original LaTeX notation
Let $n>1$ be a positive integer. Each unit square in an $n \times n$ grid of squares is colored either black or white, such that the following conditions hold: - Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge; - Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge; - Any $2 \times 2$ subgrid contains at least one square of each color. Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of $n$).

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Source and history

task_id
023666b5-2b19-54df-bda8-8f880b114cfc
task_key
test--023666b5-2b19-54df-bda8-8f880b114cfc
task_revision_id
1
config / split
/ test

Official source

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