A balloon net discovering improved solutions in one of unsolved problems in geometry: a problem of spreading points in a unit square

A balloon net model is introduced and demonstrated for discovering improved solutions in one of unsolved problems in geometry which is referred to as a problem of "spreading points in a square". How should n points be arranged in a unit square so that the minimum distance between them is the greatest? Note that d(n) is the greatest possible minimum distance between n points in a unit square. Exact results are known for n/spl les/9 and n=14, 16, 25, and 36. Many investigators including Schaer, Meir, Kirchner, Wengerodt, Goldberg, Schluter, Valette and others have studied this geometrical problem for many years. The best known result is summarized in the book of "Unsolved Problems in Geometry" (H.T. Croft, K.J. Falconer and R.K. Guy (1991)). We have found the improved solutions for n=13 and n=15 by using the proposed algorithm.