ch3cooh: (Ball Pit)
[personal profile] ch3cooh
Challenge: write up your initial instinct, then try to prove it and let me know what happens. :D
I really like this puzzle.  And I'm sure that there's a simpler proof than the one I'm using right now.

Puzzle:
GeometryArea


Solution:
The orange is exactly the same area as the sum of the two blue areas.

Proof:
Step 1: The Orange circle is exactly 1/4 of the big circle.  Proof by picture:
Picture2
(Draw another equlateral triangle inscribed in the circle, with a 180 degree rotation relative to the original triangle.  These two triangles overlap in a regular hexagon that can be split into 6 triangles like a pie.  Each of these triangles is the same size as any of the 6 points of the star now inscribed in the circle.  So it is clear that the radius of the orange circle is half the radius of the large circle.  Since A = pi*r^2, this means that the area of the orange circle is 1/4 the area of the whole circle.

Step 2: Blue area is exactly 1/3 of the remaining area in the large circle that is not part of the orange circle.  Proof by picture:
Picture1
The remaining area has three semetrical components.

Step 3: Since the orange circle is 1/4 of the large circle, the remaining interior of the large circle is 3/4 of the area.  The blue area is 1/3 of that, and 1/3 of 3/4 is 1/4.  So both the orange region and the combined two blue regions are 1/4 the area of the whole circle.
Boom!  Done!

Using this to draw a very pretty infinite summation picture. :)
Spiral of quarters 2
1/4 + 1/16 + 1/64 + ... 1/4^n  as n approaches infinity = 1/3
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