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As luck would have it, a friend and I got into a discussion on pies, and we decided to settle it with a bake-off. When we went to compare pies, I couldn't help but notice his pi are square! Having taken no part in such blasphemy, I proudly presented my traditional round pies. This of course led to further discussion and discourse as to which pies are better, and which pie actually has more. Lo and behold, one of my pies fit perfectly inside of one of his pies, touching all edges. It was obvious that his pie was bigger. But his other pie was much smaller! He must have used 2 very different pans, as his 2nd pie fit perfectly inside of my round pie. We had to take some measurements. A typical 9 inch pie has a circumference of ~28.274 inches, but, measuring very precisely, my 1st pie circumference was a measly 18.64768756. Either I used a small dish, or I measured in some arbitrary pie units. I'm not sure which. Having lost the battle of which is bigger on the 1st pie, I eagerly moved on to the 2nd pie, on which I would clearly be declared the victor. Ha! One side of his pie was a mere 3.239977414 pie units. This means that my 2nd pie is definitely smaller than my first pie though; not sure what I did there. I'm not a baker. :-P I'm not sure if we ever figured out the answer to our original questions; we were distracted by pie. Or was it pi? I know \(2\pi = 1 \) pie. Or something like that.
As this is a pi day cache, use the date for all of your pi calculations; simply 2 digits after the decimal. This will drive anybody that pays attention to significant digits nuts, but it also rather diabolically prevents the usage of online calculator tools, as they'll all use a longer version of pi. Keep the rest of your digits in your calculator as you go through the steps; it should be pretty clear where you can start dropping excess digits.
Potentially useful items:
\(E=mc^2\) \(C=\pi d\) \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) \(A = \pi r^2\) \(V = \frac{4}{3}\pi r^3\) \(a^2 + b^2 = c^2\) \(A=\pi r \times s\) \(s, s, s\sqrt2\)
You can validate your puzzle solution with certitude.