PACKING UP TO 50 EQUAL CIRCLES IN A SQUARE

The problem of maximizing the radius of n equal circles that can be pa cked into a given square is a well-known geometrical problem. An equiv alent problem is to find the largest distance d, such that n points ca n be placed into the square with all mutual distances at least d. Rece ntly, all optimal packings of at most 20 circles in a square were exac tly determined. In this paper, computational methods to find good pack ings of more than 20 circles are discussed, The best packings found wi th up to 50 circles are displayed. A new packing of 49 circles settles the proof that when n is a square number, the best packing is the squ are lattice exactly when n less than or equal to 36.