The following results were obtained by manually seeding starting positions into an optimizer, then manually fine tuning the optimizer results to account for obvious symmetries and to reduce optimizer noise. A useful strategy for the asymmetric cases in this problem is to start with a plausible solution centered on the origin and then move the entire group of cups around to maximize the total probability covered.

Note that n=5 and n=8 are offset from center, moving the asymmetrical perimeter cups towards the peak of the probability distribution to maximize total probability. With n >7 it gets harder to increase probability significantly and it seems likely that the optimal cup locations will tile the plane in center-hexagonal formation with increasingly small offsets from center, with offsets approaching zero as n becomes large.

**Note: I have updated my answer for n=6 after seeing better answers from Tom Keith and Jenny Mitchell – thanks!**

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What optimizer did you use? Care to share the code?

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Hi – it’s just my own python optimizer moving the (x,y) offsets of all the center points around the same amount after seeding center points of likely configurations . I looked at modifying a circle packing optimizer like circifly, but that’s not necessary here.

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