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Bowling ball

Q: ??? Given the positions of the finger holes in a (spherical) wooden bowling ball, can we embed lumps of lead to make the ball fair? How unique is this solution?

"Fair" presumably means at least that

1. the centre of mass is at the centre of the sphere;
2. the moment of inertia is independent of the axis we choose.

Is this sufficient? Can we do even this?

In the 2D case is condition 1 enough?

A: Condition 1: If we can find the vertices of a tetrahedron containing the centre of the ball, then we can move the centre of mass to the centre of the ball by embedding masses at these vertices.