Four Quarters
Start with a square. Divide it into four equal square quarters. In the top left corner, draw the largest possible circle. Shade it in. Divide the three remaining quarters into four quarters. In the top left corner of each of those, draw the largest possible circle. Shade it in. Continue this process of division into quarters, and shading the top left incircle. What fraction of the square will be shaded? What is the significance of this number?
-Shpilo
-Shpilo
Labels: SharedPuzzle
posted by ToM at
8:16 PM
9 Comments:
wow good question,Shpilo
the process seems never ending
wild guess-can the Fibonacci number enlighten us?
if not,then i can't touch this
Let the square be unit square. The shaded area,
A = (1/4)(π/4) + (3/4)A => A = π/4.
This is the same as the area that would be shaded if you simply drew the largest circle into the original square.
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A slightly different take. The three similar squares are simply
quarter scaled (by area) copies of the outermost square. As there
are 3 of them they account for 3/4 of the total area. So the the
odd top left square must alo be 1/4 of total area. Doubling the
top left circle (linearly), would make it fill the entire unit
sqaure and so have a diameter = 1. As the area of a circle of
diameter 1 is π/4 - that is also the same area as the original
shaded area.
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I don't think Chris understood the question. After you draw the three quarter circles, you divide all remaining empty squares into four, and draw nine circles into all the top left squares. Repeat the process, draw 81 even smaller circles, ad infinitum.
The answer is that the shaded area is the same as if you had drawn the largest possible circle in the original square, so Chris did get it right.
Shpilo. I recommend that you set up a proper account with ToM. Then you can delete the abusive comments. It's easy to do. The only way I know how to do it requires getting a free googlemail account first.
I also recommend that you do not try to reason with Anonymous. If he is a troll, then he may well be pretending to not understand the question.
I see that ToM has deleted Anonymous's abusive follow up already.
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