What the hell does "1#" and "2#" mean? Six globes, six weights. Ah, "lbs".
Fix (any) one of the weights at (0,1) -- call it W0. (radius is immaterial -- assume all weights are point sources equidistant from a center at (0,0) -- for easy math!)
For each permutation of the remaining 5, assign the weights to (0.866,0.5), (0.866,-0.5), (-0.866,0.5), (-0.866,-0.5), and (0,-1).
Compute the center of mass as ( sum (Xi * Wi), sum (Yi * Wi) ) = (X,Y) ignoring the constant scale factor (but don't forget W0 @ (0,1)!)
[Wi being weight of i-th globe, (Xi,Yi) being the above locations]
Score this as X^2 + Y^2 (no need to take sqrt) to determine it's "off-centerness"
Minimize "score" over the set of possible permutations.
[Actually, this does twice as many evaluations as needed but a no-brainer to code in this incarnation!]
High school programming assignment. (Have I missed something obvious?)
Maybe Sparky Larkin can impress you with his POWER BASIC prowess!
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Probably got answered already, but two ways...use octave and 'play' with formulas, or...wade in.
Just like packing a box, start with the largest and work your way to the smallest.
Ah, ha! I see the problem now. There's a vector torque in there. This requires a 2D distribution solution. So then think in terms of trying to 'uniformly' distribute the 'mass' over the 'area'. Thinnk largest to smallest single then in pairs, then in 'sheets' of 3 etc. Wow, tougher then it first seemed! Screw it, go buy bulbs all the same weight!
You'll have to put an unobtrusive 'ring' wieght around each bulb. Each weight cosmetically the same, yet each weight has a 'different weight to make each 'bulb system' the same weight.
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