```! Maximize the area of a polygon having n points such that the distance between any pair is <= 1. Based on an example of D. Gay. Notes: 1) This does not mean that the polygon will fit inside a circle of diameter 1, 2) For n odd, there may be multiple local optima. For; ! Keywords: Polygon; SETS: point: rho, theta; ENDSETSDATA: ! Best known solutions: n Area 3 0.4330127 4 0.5 5 0.657164 6 0.674981; ! Note, the area of a unit diameter circle is 0.7853982; point = 1..5; ENDDATA n = @SIZE(point); ! Partition the polygon into triangles, each with an apex at point n and maximize the sum of their areas; MAX = .5*@SUM( point(i) | i #gt# 1: rho(i)*rho(i-1)*@SIN(theta(i)-theta(i-1)); ); @FOR( point(i): rho(i) <= 1; @BND( 0, theta(i), @PI()); ); ! Points i and j can be at most 1 apart; @FOR( point(i)| i #lt# n: @FOR( point(j)| j #gt# i: [Dist] rho(i)^2 + rho(j)^2 - 2*rho(i)*rho(j)*@COS(theta(j) - theta(i)) <= 1 ); ); ! They must be ordered; @FOR( point(i) | i #gt# 1: theta(i) >= theta(i-1) ); ! Kill some symmetry; theta(n) = @PI(); rho(n) = 0; ```