Which carbon(s) in cyclohexene have the highest electron density? I think that carbons with highest electron density will be both $\ce C1$ and $\ce C2$ because of the loosely held $\pi$-bond between them, but according to the answer given the carbons with most electron density should be both $\ce C3$ and $\ce C6$.

How is this possible?
:-( The book gives an answer, not necessarily the right one.

Looking at the $\ce{^13C}$ NMR of cyclohexene the carbons at positions 4 and 5, in your diagram, are shifted the least and therefore have the most shielding. Thus those carbon atoms also have the greatest electron density around them.