For ten points, give the number of vertices of a cube.
Howe's theorem concerns polytopes having this many vertices. Degen's identity guarantees that the product of two sums of this many squares can be written as a sum of this many squares. The  smooth  version of a polygon with this many sides is conjectured to be the least e ciently packable centrally symmetric shape, and any cubic curve passing through this many intersection points of two other cubic curves must pass through all such intersection points. Given any three circles, there are this many circles in the plane to which those three are all   tangent, and a regular polygon with side length x and this many sides has area (2 + 2√2)x2. For ten points, give the number of vertices of a cube.
Howe's theorem concerns polytopes having this many vertices.
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