What is the sum of the squares of all real values of $x$ for which $|2-|x| |=1$?
I notice that the equation involves absolute values, which are always non-negative, so I can split it into two cases: either $2-|x| = 1$ or $2-|x| = -1$. In the first case, $2-|x| = 1$, I can isolate $|x|$ by subtracting $1$ from both sides, and get $|x| = 1$. This means that $x$ can be either $1$ or $-1$, since both of them have an absolute value of $1$. In the second case, $2-|x| = -1$, I can isolate $|x|$ by adding $1$ to both sides, and get $|x| = 3$. This means that $x$ can be either $3$ or $-3$, since both of them have an absolute value of $3$. Therefore, the possible values of $x$ are $1, -1, 3, -3$. To find the sum of their squares, I need to evaluate $1^2 + (-1)^2 + 3^2 + (-3)^2$. This is equal to $1 + 1 + 9 + 9$, which is $20$.

