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I was reviewing past exam questions for a math competition known as UIL Calculator Applications when I stumbled across the following question:

A large amount of dough is rolled out and as many circular cookies as possible are cut from the rolled-out dough. The remaining dough is piled together, rerolled and more circular cookies are similarly cut. What percent of the original amount of dough is left over?

Here's my incorrect attempt, if you'd like:

First, I imagined that I was cutting one circle out of a square with side length $x$, with 4 points tangent to the circle. Therefore, the remaining area would be $$x^2-\pi(\frac{x}2)^2$$

Then, I imagined that I was cutting nine circles out of a square with side length $x$, with each circle tangent to the square and/or to adjacent circles. Therefore, the remaining area would be $$x^2-9\pi(\frac{x}6)^2$$ which is the same result as the first scenario. So the percent of original amount of dough left over after 1 "cut" for any number of circles is $$\frac{x^2(1-\frac{\pi}4)}{x^2}\approx21.46\%$$ and after 2 "cuts" the percent of original amount of dough left over is $$(21.46\%)^2\approx4.61\%$$

According to the answer key, however, the answer rounded to 3 significant digits is $.867\%$ and I'm not sure as to how.

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A grid of hexagons is more efficient than a grid of squares, using $\frac{\pi}{2\sqrt{3}}$ of the dough instead of $\frac{\pi}{4}$.

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