9.56 On a rectangular piece of paper, there are 16 black points distributed. For each pair of points, the following operation is performed: connect these two points with a line segment, and use this line segment as the diagonal to construct a rectangle whose sides are parallel to the edges of the rectangular paper, then color this rectangle red (note that black can show through the red). How many edges does the resulting figure have (provide all possible answers depending on the different distributions of the points)?
[Solution] Place the rectangular paper upright and select the topmost, bottommost, leftmost, and rightmost points from the 16 black points (if two points are the same, choose one of them). Then, use these 4 points as vertices to form a convex quadrilateral $ABCD$. Construct red rectangles with $AB, BC, CD, DA$ as diagonals, respectively. The 4 red rectangles will exactly form a large red rectangle, which is the rectangle enclosed by the horizontal lines through the top and bottom points and the vertical lines through the left and right points. Clearly, all other black points are inside or on the boundary of this large red rectangle. Therefore, the red rectangles with the other points as diagonals are all covered by this large red rectangle, so the red figure drawn is a rectangle with 4 sides.

We point out that when the convex hull of the 16 black points is a triangle or a slanted line segment, among the topmost, bottommost, leftmost, and rightmost 4 points selected, two points can coincide, or two pairs of points can coincide, but the conclusion is the same, i.e., the resulting figure has 4 sides. When the 16 black points are on a horizontal or vertical line, the resulting rectangle degenerates into a line segment, of course, with only 1 side.