[Big-Daddy]

Let's take your attempts one at a time:

⁺N=N-N=N-N

First, your charges are wrong.  Nitrogen with only two bonds is formally -1, not +1.  (Do your formal charge calculation when in doubt, but you should learn to recognize it quickly.  Three-bonded nitrogen is neutral, two-bonded nitrogen is -1 and four-bonded nitrogen is +1.  Easy way to remember is that dinitrogen, a neutral compound, is triple bonded and the cations in ammonium salts are NH₄⁺.)  The singly bonded nitrogen on the right hand side would formally be -2, not 0 (so the overall charge of this molecule would be -1, not +1 as required by the problem), which brings me to the second problem: in some cases you'll see a nitrogen with a -1 formal charge (doubly bonded) but almost never -2 (singly bonded).  So this is what I meant by: learn the conventional bonding patterns of common elements.  Nitrogen can commonly accommodate 2, 3 or 4 total bonds.  Your resonance structures should not include nitrogens that are bonded only once or five times, shouldn't include oxygens that are bonded four times and carbons pretty much should always be bonded 4 times (other versions exist but they're for the most part unstable so you shouldn't worry about them). 

N-N=N⁺=N-N

Nitrogens on the ends would each be formally -2 charge (for an overall molecular charge of -3) and as pointed out above, nitrogen shouldn't be bonded only once.

N=N-N⁺-N=N

Nitrogens on the ends would each be formally -1 charge and the nitrogen you've identified as having +1 charge also has a -1 charge (for an overall molecular charge of -3).

Haha sorry, it was into the morning's early hours and I forgot for a second that N cannot disobey the octet rule! My plan with giving N just 1 bond was that this involves its 5th electron, leaving 2 lone pairs, but then the total is only 6 :p

Should I always assume that if N has 4 bonds it requires a positive formal charge, if it has 3 it is neutral, if it has 2 it has a negative formal charge? (And so forth, i.e. 5 bonds -> +2, 1 bond -> -2 formal charges). I know it is always the case with O that 2 bonds mean O is neutral, 3 mean it is positive and 1 means it is negative. (Any exceptions to this? O's pretty important) If C has 5 it will be -1, if it has 4 it will be neutral, if it has 3 it will be +1, 2 will be -2, etc. So I suppose, is there are systematic way to work this out, which extends maybe to the transition metals? I am familiar with most of the main elements but it would make it far easier to find the likely isomers if I understood what formal charge each atom will take up given a certain number of bonds. Prior to this discussion I think the only one I was crystal clear on was oxygen (as it comes up so often in examples).

[Big-Daddy]

By the way, you'll notice that all the "correct" isomers have the same total number of bonds - 8.  Because the relationship between number of bonds to a nitrogen and it's formal charge is always constant, I got to thinking that for a given number of nitrogens connected in a linear fashion, there should be a constant relationship between the total number of bonds in the molecule, the number of nitrogens in the chain, and the overall formal charge.  I played around with the math and I came up with a formula.

The total number of bonds, D(n) in a linear chain of n nitrogens with overall formal charge C is

[tex]D(n) = \frac{C + 3n}{2}[/tex]

I tested it out on a bunch of test cases and it seems to work.  (In the case under consideration, n = 5 and C = +1, so the total number of bonds is 8.)  Since D(n) has to be a whole number, any charge/n combination that gives a fraction is impossible.  For instance, a chain of five nitrogens with a +2 charge doesn't work (D = 8.5), so there's no way to make a chain of nitrogens bond in such a way that the formal charge is +2, and playing around with the structures confirmed it as far as I can tell.  I could probably modify the formula for branched chains and also ring structures and such, I suppose, but didn't get that far and the math would get more complicated in a hurry. 

Anyway.... yeah, I'm a nerd but I thought it was kind of interesting.

This formula deserves far more time than I have for it at the moment. After my exams finish I will return to talk about this a bit more.

[Big-Daddy]

Haha sorry, it was into the morning's early hours and I forgot for a second that N cannot disobey the octet rule! My plan with giving N just 1 bond was that this involves its 5th electron, leaving 2 lone pairs, but then the total is only 6 :p

Should I always assume that if N has 4 bonds it requires a positive formal charge, if it has 3 it is neutral, if it has 2 it has a negative formal charge? (And so forth, i.e. 5 bonds -> +2, 1 bond -> -2 formal charges). I know it is always the case with O that 2 bonds mean O is neutral, 3 mean it is positive and 1 means it is negative. (Any exceptions to this? O's pretty important) If C has 5 it will be -1, if it has 4 it will be neutral, if it has 3 it will be +1, 2 will be -2, etc. So I suppose, is there are systematic way to work this out, which extends maybe to the transition metals? I am familiar with most of the main elements but it would make it far easier to find the likely isomers if I understood what formal charge each atom will take up given a certain number of bonds. Prior to this discussion I think the only one I was crystal clear on was oxygen (as it comes up so often in examples).

I've just found examples where this isn't the case, for instance in NO, nitrogen has 2 bonds but is not negative (it has a free radical instead). Which leaves me confused as to what sort of guidelines to apply ...

Help with this would be much appreciated

[Corribus]

Yes, well, free radicals aren't typically stable molecules because they react vigorously in redox reactions.  NO, for example, is rapidly oxidized in air.  Hydroxyl radical is a singly bonded oxygen and has a formal charge of zero and also rapidly reacts even in the gas phase.  And so on.  Small gas molecules are more likely to be encountered as free radicals, so they're the most likely place to see these kinds of examples.

There are exceptions to every rule.  The best way is to learn the most usual way molecules behave and then understand when and where and why there are exceptions. 

[Big-Daddy]

Yes, well, free radicals aren't typically stable molecules because they react vigorously in redox reactions.  NO, for example, is rapidly oxidized in air.  Hydroxyl radical is a singly bonded oxygen and has a formal charge of zero and also rapidly reacts even in the gas phase.  And so on.  Small gas molecules are more likely to be encountered as free radicals, so they're the most likely place to see these kinds of examples.

There are exceptions to every rule.  The best way is to learn the most usual way molecules behave and then understand when and where and why there are exceptions.

OK so I will proceed with the assumption that the guidelines I wrote for N, O and C are all generally good. Later I may come up with a generalized form and come to ask you.

So why does N end up with a free radical so often? e.g. in NO₂, I'm guessing both O atoms are going to require 2 bonds to keep them neutral, and that leaves the N with just 1 electron left - a free radical.