The Solution
The trick is to think of a series of regular polygons with weights placed at the vertices—equilateral triangle, square, pentagon, hexagon, and septagon. Adding the missing weights (blue dots) will create equilibrium about the center point because of the symmetrical distribution of the weights.
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Science is reinventing play, from extreme sports to gamification to ridiculous roller coasters to the playgrounds of tomorrow, and this issue is chock full of fun. Also, on a less fun note: Did global warming destroy my hometown?