[TheWorstPlayer]

This is not really elegant. It is effective, but it is a ton of work and writing, etc. Not elegant at all. A more elegant solution that I came up with is to weigh 4 against 4. If it doesn't balance, move three normal ones (which weren't involved in the first weighing) to the heavy side, three from the heavy side to the light side and three from the light side off. Now weigh again.

If the heavy side is still heavy, then either the one heavy left on it is heavy or the one light on the light side is light. Weigh the light one against a normal and if it is light, it is that one that is weird and it is light. If they balance, then the heavy one from the heavy side is the weird one and it is heavy.

If the heavy side from the first weighing is now light on the second weighing, then you know that one of the three heavy ones that you moved to the original light side must be heavy. So weigh one against one and if it is unbalanced it is the heavy one and it is heavy. If it is balanced, the one you didnt weigh is weird and it is heavy.

If the second weighing balances, then one of the three light ones that you took off is light. Repeat the three heavies procedure to find the light one.

If the initial weighing balanced, then you know that it is one of the four that you didn't weigh that is weird. For the second weighing, weigh three of those against three normal ones. If the weird side is heavy, repeat the heavy procedure from above to determine which one it is and it is heavy. If the weird side is light, same procedure to determine which one and it is light. If they balance, then the one you didn't weigh at all yet is weird. Weigh it against a normal one to see if it is light or heavy.

QED.