$n \times n$ square paint

Probability Level 2

A board of $n cm \times n cm$ is divided in $n^2$ little squares of $1cm \times 1 cm$ . If each square can be painted black or white. Find all ways to color the board such that each square of $2 cm \times 2 cm$ formed by $4$ little squares with a common vertex has $2$ black squares and $2$ whitesquares.

2^{n+1}

2^{n+1}-1

2^n

2^{n+1}-2

1 solution

Paola Ramírez
Jan 4, 2015

First column can be painted of $2^n$ .

There are two ways to paint the first column alternating black and white squares, so the next columns, each one, can be painted of to ways $\therefore$ exist $2 \times 2^{n-1}$ colorations.

If first column has to equal colors together next column coloration will be defined $\therefore$ exist $2^n-2$ colorations.

Total colorations are

$2 \times 2^{n-1} +2^n-2=2 \times 2^n-2=\boxed {2^{n+1}-2}$

n × n n \times n n × n square paint

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$n \times n$ square paint