There is a subsquare centered at the center of square and all it's four vertices are white.
I'm not sure if this is supposed to mean that the center of the subsquare is also white. Am I wrong?
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The vertices of a square are the corners, not the entire square. We don't know the color of the space inside the lines of the square.
In the image below, the vertices are the points A, B, C, and D. In this case the vertices appear black, while the square itself is (mostly) gray.
Other that the vertices mentioned, we have no information about the color of any of the other objects.
Side note: In your sentence, it should be its not it's.
... and all its vertices are white.
Its (without the apostrophe) is the possessive. It's is a contraction for "it is". Don't feel bad, as many native speakers frequently make this mistake.
I like Andrew's answer. Vertices are single points and thus do not have area, so it's hard to conceptualize a vertex being colored in with something. A couple asides:
Saying the specific number of vertices is redundant since a square is well-defined to have four vertices. I would only include the specific number of vertices if I wasn't sure if the person/people I'm communicating with didn't know that information. For example, when talking to young children but certainly not to educated adults.
"Centered at the center" is redundant and thus a bit confusing since there are extra words. Instead of both, use only either one of them.
So, we're left with:
There is a subsquare centered in another square and all its vertices are white
There is a subsquare at the center of another square and all its vertices are white