Catalogue of arrangements for A336782 Arrangements of n squares with side lengths s = 1, 2, ..., n such that the number of 1 X 1 cells formed by the boundaries of all squares is maximized. All squares shall lie within the largest square. Exhaustive enumeration by Hugo Pfoertner in September 2020. Number of essentially distinct solutions: A336782(n). Number of 1X1 cells: A336659(n). The position of a square is given by the coordinate pair (x,y) of the lower left vertex. Squares are noted in descending order of size. Arrangements which differ only in the position (x1,y1) are called 1-siblings. Coordinates of squares n, n-1, ..., 2 are given only for the first 1-sibling. This list serves as a catalogue for the illustrations. Only the first 1-sibling is shown in full. Each position (x1,y1) is marked by a little cross (+). n = 1, 1 solutions, 1X1 cells: 1 x1 y1 0 0 n = 2, 1 solutions, 1X1 cells: 1 x2 y2 x1 y1 0 0 0 0 n = 3, 1 solutions, 1X1 cells: 2 x3 y3 x2 y2 x1 y1 0 0 0 0 1 2 n = 4, 1 solutions, 1X1 cells: 4 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 2 2 3 3 n = 5, 1 solutions, 1X1 cells: 8 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 2 2 1 3 3 2 n = 6, 16 solutions, 1X1 cells: 10 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 2 2 3 1 4 3 3 0 0 0 0 0 2 3 1 2 1 4 2 .. .. .. .. .. .. .. .. .. .. 5 2 .. .. .. .. .. .. .. .. .. .. 5 5 0 0 0 0 0 2 3 3 3 2 1 5 .. .. .. .. .. .. .. .. .. .. 5 1 0 0 0 0 1 2 0 3 2 4 1 3 0 0 0 0 1 2 3 1 2 1 4 3 .. .. .. .. .. .. .. .. .. .. 5 2 0 0 0 0 1 2 3 1 4 3 3 2 0 0 0 0 1 2 3 3 2 4 4 3 .. .. .. .. .. .. .. .. .. .. 5 1 .. .. .. .. .. .. .. .. .. .. 5 4 0 0 0 0 2 2 1 3 1 2 3 4 0 0 0 0 2 2 1 3 1 4 3 3 0 0 0 0 2 2 1 3 3 4 2 3 n = 7, 5 solutions, 1X1 cells: 15 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 0 2 3 1 4 3 5 5 1 .. .. .. .. .. .. .. .. .. .. .. .. 5 4 0 0 0 0 0 2 3 3 4 1 4 5 3 4 0 0 0 0 2 2 1 3 4 1 5 1 1 4 .. .. .. .. .. .. .. .. .. .. .. .. 3 6 n = 8, 1 solutions, 1X1 cells: 22 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 0 1 3 4 2 5 1 6 3 7 7 n = 9, 1 solutions, 1X1 cells: 28 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 2 2 1 3 4 0 5 0 3 1 7 1 6 0 n = 10, 4 solutions, 1X1 cells: 34 x10 y10 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 0 3 3 1 4 5 0 6 2 7 1 2 8 5 8 ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5 9 0 0 0 0 0 0 3 3 1 4 5 5 6 2 7 1 2 8 4 3 0 0 0 0 0 0 3 3 1 4 5 5 6 2 7 1 4 8 4 3 n = 11, 12 solutions, 1X1 cells: 41 x11 y11 x10 y10 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 0 1 3 4 4 2 5 6 1 7 3 8 2 3 9 6 9 ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6 10 ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9 8 0 0 0 0 0 2 0 3 4 1 5 1 6 6 7 5 3 1 5 9 1 2 ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 10 0 0 0 0 1 2 0 1 2 4 5 0 6 0 7 3 4 0 9 2 5 5 ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9 8 0 0 0 0 1 2 3 1 2 4 0 0 0 0 0 3 4 0 0 2 0 8 ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1 8 ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5 5 ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 0 ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 1 n = 12, 8 solutions, 1X1 cells: 52 x12 y12 x11 y11 x10 y10 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 0 0 0 4 4 5 5 1 6 3 7 2 8 8 3 6 6 6 5 ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 7 ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9 1 ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1 0 0 0 0 0 0 0 0 4 4 5 5 2 6 1 7 3 8 8 3 1 6 6 5 ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 7 ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9 1 ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1 n = 13, 24 solutions, 1X1 cells: 60 x13 y13 x12 y12 x11 y11 x10 y10 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 0 8 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1 8 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 8 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 9 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4 4 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6 5 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8 4 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9 7 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 1 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 2 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 6 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10 9 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 1 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 2 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 6 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 11 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 11 12 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 12 1 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 12 2 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 12 6 0 0 0 0 0 0 0 0 4 4 5 5 2 6 1 7 3 8 6 9 3 6 10 1 1 8 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 8 0 0 0 0 0 0 0 0 4 4 5 5 2 6 1 7 3 8 6 9 3 6 11 1 1 8 ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2 8 n = 14, 2 solutions, 1X1 cells: 70 x14 y14 x13 y13 x12 y12 x11 y11 x10 y10 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 0 1 3 0 4 2 5 6 1 7 0 8 2 9 6 5 2 9 10 12 10 10 2 ... ... ... ... ... ... ... ... ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 13 13 n = 15, 1 solutions, 1X1 cells: 83 x15 y15 x14 y14 x13 y13 x12 y12 x11 y11 x10 y10 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 0 0 0 0 4 4 1 5 3 6 2 7 8 3 9 2 10 2 11 1 12 6 10 9 12 11 >