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Omni-MATH / Let n be a positive integer. We are given a 3n×3n board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal i…

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problem

Let nn be a positive integer. We are given a 3n×3n3n \times 3n board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a 2×22 \times 2 square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all nn for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.
Plain-text mathematical notation (without MathML)
Let n be a positive integer. We are given a 3n×3n board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a 2×2 square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all n for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.
Original LaTeX notation
Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.

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Source and history

task_id
5488c94a-8f24-5e5f-89ff-f23c647a3597
task_key
test--5488c94a-8f24-5e5f-89ff-f23c647a3597
task_revision_id
1
config / split
/ test

Official source

initial import