Question 1162655: A three by three grid is divided into nine unit squares. The top left unit square is shaded. Each of the other eight unit squares is either shaded or unshaded. How many such grids do not contain a two by two square in which each of its four unit squares is unshaded?
Answer by greenestamps(13198)   (Show Source):
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The top left unit square is shaded.
The other 8 unit squares are either shaded or unshaded. That is 2 choices for each of the 8 squares, for a total of 2^8=256 different grids with the top left unit square shaded.
With the top left unit square shaded, there are 3 sets of four unshaded unit squares that form a 2x2 square.
For each of those three sets of four unit squares, with the top left unit square shaded there are four unit squares left that are either shaded or unshaded. That is 2^4=16 possible combinations.
So, with the top left unit square shaded, there are 3*16=48 grids which DO contain an unshaded 2x2 square.
And so the number of grids with the top left unit square shaded that do NOT contain an unshaded 2x2 square is 256-48 = 208.
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