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In this paper, we prove that if the solution of the planar N + 1 -body problem has N bodies rotating at the vertices of a regular N -gon then the N bodies have equal mass and the remaining body must be at the center. [ABSTRACT FROM AUTHOR]

The popular puzzle on finding the length of the side of the largest square which can be enclosed in a given triangle, is thought to have had its origins in the mathematical studies of Heron in the 1st century A.D. It is believed by some that he used the 'link' between geometry and what would become known as algebra to find a solution to this puzzle. Many people have since studied this puzzle and its inverse, the puzzle of finding a minimum area triangle for a given square, but few present the entire solution set. We study the inverse puzzle for the general case of a given regular n-gon. A new geometry method will be presented which will assist in the solution of this puzzle. [ABSTRACT FROM AUTHOR]