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Here is my sphere:

In the Edit mode

I want to subdivide the face I'm selecting into four, like I did with the other ones. But, when I go to W->Subdivide Smooth, the surrounding dots just move closer to the face, and the face doesn't subdivide in four. Why is this and how can I fix it?

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the faces you're subdividing are NGons, not quads or tris, because you subdivided the surrouding faces. Subdividing them adds vertices but no edge.
NGon problem

What you can do is dissolving the non desired vertices with X
dissolve

Then joining them to make edge and faces, select 2 vertices and hit J
join

subdivide smooth on the new edge and join
subd join

final join

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