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The difference between fitting and matching
Two things match if they resemble each other, for example, two socks or two cups from a set of tableware. Two things fit if they are closely linked without matching; for example, a key fits a lock, a hat fits a head, or a name fits a face.
It is convenient to use both words, but I think fitting is actually just a special kind of matching, isn't it? If a hat fits, the inside of the hat-band matches the outside of your head (as regards length); or if a key fits a lock, the kinky profile you see on the key matches a corresponding profile somewhere in the lock's interior --- you just don't see this second part.
When an answer fits a problem, the situation is a bit more abstract but the same idea can be extended. In some sense, describing the problem describes a space that needs to be filled, and if something fills that space, it is an answer to the problem. We don't need to go into all the details; I just want to suggest that fitting and matching are rather similar. Then when we come to ask, "What process runs inside your brain when a match is noticed?" we can suppose that pretty much the same process is provoked by a fit; we don't have to start a whole new search --- one brain-process works for both.
(On the Tree, the first six twigs at lower left are examples of fitting, and all the rest are examples of matching. Different kinds of matching are separated out by looking at the Tree's branches.)