Electron Shell Configuration Is there a way to predict the electron configuration of an element, for example 
Copper is 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1 3d^10  ?
 A: In theory, yes: If you assume a point-like positive nucleus and use Feynman's QED theory, you can predict the electron configuration of any element. Feynman even gives some rough examples in his Lectures on Physics, which now are online (yahoo!... as in yippee!, versus a certain web site). Since this is a homework question, I heartily recommend looking up his approximate discussion of the topic.
In practice, however, any element as complex as one of the transition elements requires both guesswork and a lot of approximations that often are informed directly or indirectly by observation of the actual elements. I am not aware of any precise predictions of something as complex as copper, which is a good example of how gnarly and weird the valences can get for some elements. Copper(III) compounds were always a favorite of mine back in high school, for example, and that was decades (sigh) before they became famous for superconductors. The fact that no one has one a Nobel prize yet for explaining how copper(III) enables non-Cooper-pair superconductivity is another indication of how difficult the computational models become for such elements.
A: Terry is correct that to be sure of the configuration requires a complex calculation. However for the vast majority of the known elements the electron configuration is correctly predicted by the Madelung rule. This gives the order in which the atomic orbitals are occupied.
As it happens copper is one of the few elements that violate this rule. The Madelung rule predicts the $4s$ orbital should be occupied before the $3d$ orbital, but as you say in your question the configuration is actually $4s^13d^{10}$. The rationalisation of this is that filling the $3d$ orbital, i.e. going from $3d^9$ to $3d^{10}$ makes a big enough jump in energy to overcome the energy lost going from a filled $4s$ orbital to $4s^1$, which is all very well but predicting this from first principles is a challenge.
