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Let \(n\) be a positive integer. We have a board of dimensions \(n\times n\), divided in \(n^2\) "houses" and we mean to paint each one of the houses in either blue or white. We say that the coloration is "rectangulable" if, between any four houses whose centres form a rectangle of parallel sides to the board's, the number of houses painted in each color is even.

Determine, in function of \(n\), the number of rectangulable paintings of the board.

Note by Tomás Carvalho
5 months, 4 weeks ago

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Suppose we have a \(6 \times 6\) board. If we have an "internal" \(3 \times 5\) rectangle with \(15\) squares, there would be no way for the number of houses painted in each color to be the same. Perhaps you will need to relax the condition so that the number of houses painted (within every internal rectangle) in each color differ by no more than \(1\). Brian Charlesworth · 5 months, 3 weeks ago

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@Brian Charlesworth When I say the number of houses painted in each color, Im refering to those 4 houses and not the ones inside the rectangle formed. ( a rectangulable painting is such that between any four houses forming a rectangle there are 0,2 or 4 painted each color. Tomás Carvalho · 5 months, 3 weeks ago

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@Tomás Carvalho O.k., great. Thanks for the clarification. :) Brian Charlesworth · 5 months, 3 weeks ago

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