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It is helpful to know all the oxidation numbers and that is one way to do it, but there are several other ways as well. We will usually be given a table of reduction reactions with there E values, so you can tell from that as well. It also helps to look at oxygen, because that is where the root of the word oxidation came from. If a molecule gains oxygen it usually means it was oxidized and if a molecule loses oxygen it usually means it has been reduced.
For the most part, with a bit of practice, it becomes easier to tell which is being oxidized and which is being reduced. We know O and H, all alkali metals, and all halogens will be unchanging for the most part, so we are looking for transition metals. Once we know the molecule with the transition metals, then yes within that molecule we must find every oxidation number just to make sure, but we don't need to find the oxidation numbers for all molecules.
You just have to find the oxidation numbers for the elements that are changing(being oxidized or reduced). If you do not know which elements are being oxidized or reduced then you should calculate all of the oxidation numbers to find out.
What I usually do is identify the ones that are unchanging like how oxygen and ozone have oxidation numbers of zero. From, there I'd find the oxidation numbers for the transition metals or whatever other element/compounds there are. Based on the changes in oxidation numbers of some species in the reactant and product side, you can identify what's being oxidized and reduced.
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