Assuming the octet rule is obeyed, how many bonds and how many lone pairs will atoms of oxygen (o) and nitrogen (n) have in order to have formal charges of +1?

The Formal charge is not an actual charge of  any atom, it is the hypothetical charge if all the bonds were 100% covalent.

Oxidation state is the hypothetical charge on an atom if all the bonds are 100% ionic. There are no 100% ionic bonds.  

Formal charge (FC) calculation  

FC = VE - NBE - ½BE        [ VE=valence electrons, NBE=non-bonding electrons, BE=bonding electrons  ]

Oxygen has six valence electrons. Oxygen can have a formal charge of +1 with two lone pairs and one bond, but that doesn't give oxygen an octet of electrons.

FC = 6 - 4 - 1 = +1  

A second possibility is one lone pair and three bonds which will give oxygen a +1 formal charge. (As in H3O^+)  

Nitrogen has five valence electrons. Nitrogen can have a formal charge of +1 with one lone pair and two bonds. Nitrogen won't have an octet, either.  

FC = 5 - 2 - 2 = +1  

Another possibility is no lone pairs and four bonds.  

FC = 5 - 0 - 4 = +1  

explanation: it gains negative electrons, therefore becoming negatively charged. as charge cannot be created, nor destroyed, only transferred, this leaves the wool positive. now that they have opposite charges, the two items attract.

2b2t

explanation:

2b2t