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bonanova

How much white?

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This looks like a Sierpinski triangle. It can be defined iteratively by removing the "middle triangle" from the figure, then the "middle triangles" of the remaining 3 triangles, then the middles of those...

 

At each step, the shaded portion is 3/4 of what it was before. So the area of the remaining shaded portion after N steps is (3/4)N. If continued ad infinitum, the limit is zero. Therefore the area of the shaded portion is zero, and the white portion must then be 1.

 

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Area of the white is = 1 (1/4) + 9(1/4 x 1/4 ) + 27 ( 1/4 x 1/4 x 1/4) + 81 ( 1/4 x 1/4 x 1/4 x 1/4 )  The lowest common multiple of all of the divisors  is 1024 or 4 x 4 x 4 x 4  

Area of the white is =  256/1024 +  192 / 1024 +144/1024 +108/1024 + 81/1024;  Totaling these =781 / 1024  then dividing 1024 into 781 you get 0,762695313 rounding off to 3 places = 0.763

 

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