if summation of square of difference between elements and their mean is given then can we find the elements?

Question

$$\sum_{i=0}^n (a_i-u)^2 = sum $$ u = mean/average of the elements

where value of sum and n is given. Can we find value of each $a_i$?

Answer 1

No. Suppose $n=2$. On the one hand, if $a_{0}=-1$ and $a_{1}=1$, then $\mu=0$ and $a_{0}^{2}+a_{1}^{2}=2$. On the other hand, if $a_{0}'=a_{1}'=0$, then $\mu'=0$ and $a_{0}'^{2}+a_{1}'^{2}=0$.

As @confused pointed out, the reason the $a_{i}$'s cannot be recovered is because you have one equation, but more than one unknown variable.