Riddle with Pi = 3 This is a riddle someone posted on Google+, so please forgive it's triviality - I'm asking here because I just can't figure out what exactly is wrong, and it really bugs me ;)
I think something is not right with the square root at the end, but I'm not sure.
Here is the post
and here the riddle - "Find the mistake":
\begin{align*}
x&=(\pi+3)/2\newline
2x&=\pi+3\newline
2x(\pi-3)&=(\pi+3)(\pi-3)\newline
2\pi x-6x&=\pi^2-9\newline
9-6x&=\pi^2-2\pi x\newline
9-6x+x^2&=\pi^2-2\pi x+x^2\newline
(3-x)^2&=(\pi-x)^2\newline
3-x&=\pi-x\newline
\pi&=3\newline
\end{align*}
 A: Your instincts were right: if you take the square root on both sides on the 7th line, you will have to keep in mind that each side can have a positive or a negative algebraic sign afterwards. As such, the 8th line should look like this:
$$\pm(3 - x) = \pm(\pi - x)$$
This results in four combinations, two of which give the wrong result of "$\pi = 3$" ($++$ and $--$), while the other two ($+-$ and $-+$) will result in the 1st line, which is correct.
A: You're quite right about where you think the error is - in fact, every other step is valid. Since $\pi > x > 3$, $3 - x$ is negative; hence, the step which involves a square root should actually read
\begin{align*}
(3 - x)^2 = (\pi - x)^2 &\implies \sqrt{(3 - x)^2} = \sqrt{(\pi - x)^2} \\
&\implies |3 - x| = |\pi - x| \\
&\implies x - 3 = \pi - x 
\end{align*}
Upon rearranging this for $x$, we simply get back the original definition that $x = (\pi + 3) / 2$.
A: Up to step 4 seems perfectly valid.  
But I don't see where exactly between steps 4 and 5 that we've established that 2πx is interchangeable with 9 on the left side.
Though we seem to then depend on that assumption on the right side as well to substitute in -2πx for the -9 we got fairly, and all the following steps all seem to continue on that interesting assumption.
