I start with a regular pentadecagon (15-sided shape) with has a kilogram weight on each of its 15 vertices. Clearly the centre of gravity will be in the exact centre of the shape.

Someone comes along with a 16th kilogram weight and places it at vertex ‘O’.

How can I take the eight weights from vertices A to H and redistribute them amongst vertices A to H so that the overall centre of gravity is once more in the exact centre of the shape? The weights cannot be subdivided, and must be placed on those vertices, not elsewhere on the shape.

Bonus question: if you wanted to put a 17th weight somewhere on the left-hand side (positions I to O), such that the eight weights on the right-hand side (A to H) could again be rearranged to balance the system, where would the 17th weight need to be placed?