Arrangements of n-1 squares with sizes (n-1) X (n-1), (n-2) X (n-2), ..., 1 X 1 inside an nXn square such that the number of 1 X 1 squares formed by the boundaries of all squares is maximized. Exhaustive enumeration by Hugo Pfoertner 2020-08-06 n = 1 .. 12 2020-09-02 n = 13, 14, 15 Number of essentially distinct solutions: A336782(n), number of 1X1 squares: A336659(n). The position of the squares is given by the coordinate pair (x,y) of the lower left vertex. By default, the nXn square and the (n-1)X(n-1) square have their lower left vertex at 0,0 and are not shown in the lists below. n = 3, 1 solution, 1X1 squares: 2 x1 y1 1 2 n = 4, 1 solution, 1X1 squares: 4 x2 y2 x1 y1 2 2 3 3 n = 5, 1 solution, 1X1 squares: 8 x3 y3 x2 y2 x1 y1 2 2 1 3 3 2 n = 6, 16 solutions, 1X1 squares: 10 x4 y4 x3 y3 x2 y2 x1 y1 0 2 2 3 1 4 3 3 0 2 3 1 2 1 4 2 0 2 3 1 2 1 5 2 0 2 3 1 2 1 5 5 0 2 3 3 3 2 1 5 0 2 3 3 3 2 5 1 1 2 0 3 2 4 1 3 1 2 3 1 2 1 4 3 1 2 3 1 2 1 5 2 1 2 3 1 4 3 3 2 1 2 3 3 2 4 4 3 1 2 3 3 2 4 5 1 1 2 3 3 2 4 5 4 2 2 1 3 1 2 3 4 2 2 1 3 1 4 3 3 2 2 1 3 3 4 2 3 n = 7, 5 solutions, 1X1 squares: 15 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 2 3 1 4 3 5 5 1 0 0 2 3 1 4 3 5 5 4 0 2 3 3 4 1 4 5 3 4 2 2 1 3 4 1 5 1 1 4 2 2 1 3 4 1 5 1 3 6 n = 8, 1 solution, 1X1 squares: 22 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 1 3 4 2 5 1 6 3 7 7 n = 9, 1 solution, 1X1 squares: 28 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 2 2 1 3 4 0 5 0 3 1 7 1 6 0 n = 10, 4 solutions, 1X1 squares: 34 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 3 3 1 4 5 0 6 2 7 1 2 8 5 8 0 0 3 3 1 4 5 0 6 2 7 1 2 8 5 9 0 0 3 3 1 4 5 5 6 2 7 1 2 8 4 3 0 0 3 3 1 4 5 5 6 2 7 1 4 8 4 3 n = 11, 12 solutions, 1X1 squares: 41 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 1 3 4 4 2 5 6 1 7 3 8 2 3 9 6 9 0 0 1 3 4 4 2 5 6 1 7 3 8 2 3 9 6 10 0 0 1 3 4 4 2 5 6 1 7 3 8 2 3 9 9 8 1 2 0 1 2 4 5 0 6 0 7 3 4 0 9 2 5 5 1 2 0 1 2 4 5 0 6 0 7 3 4 0 9 2 9 8 1 2 3 1 2 4 0 0 0 0 0 3 4 0 0 2 0 8 1 2 3 1 2 4 0 0 0 0 0 3 4 0 0 2 1 8 1 2 3 1 2 4 0 0 0 0 0 3 4 0 0 2 5 5 1 2 3 1 2 4 0 0 0 0 0 3 4 0 0 2 8 0 1 2 3 1 2 4 0 0 0 0 0 3 4 0 0 2 8 1 0 2 0 3 4 1 5 1 6 6 7 5 3 1 5 9 1 2 0 2 0 3 4 1 5 1 6 6 7 5 3 1 5 9 10 10 n = 12, 8 solutions, 1X1 squares: 52 x10 y10 x9 y9 x8 y8 x7 y7 x6 y6 x5 y5 x4 y4 x3 y3 x2 y2 x1 y1 0 0 0 0 4 4 5 5 1 6 3 7 2 8 8 3 6 6 6 5 0 0 0 0 4 4 5 5 1 6 3 7 2 8 8 3 6 6 8 7 0 0 0 0 4 4 5 5 1 6 3 7 2 8 8 3 6 6 9 1 0 0 0 0 4 4 5 5 1 6 3 7 2 8 8 3 6 6 10 1 0 0 0 0 4 4 5 5 2 6 1 7 3 8 8 3 1 6 6 5 0 0 0 0 4 4 5 5 2 6 1 7 3 8 8 3 1 6 8 7 0 0 0 0 4 4 5 5 2 6 1 7 3 8 8 3 1 6 9 1 0 0 0 0 4 4 5 5 2 6 1 7 3 8 8 3 1 6 10 1 n = 13, 24 solutions, 1X1 squares: 60 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 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 0 8 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 1 8 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 2 8 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 2 9 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 4 4 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 6 5 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 8 4 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 9 7 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 10 1 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 10 2 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 10 6 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 10 9 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 11 1 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 11 2 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 11 6 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 11 11 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 11 12 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 12 1 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 12 2 0 0 0 0 4 4 5 5 1 6 3 7 2 8 6 9 3 6 7 6 12 6 0 0 0 0 4 4 5 5 2 6 1 7 3 8 6 9 3 6 10 1 1 8 0 0 0 0 4 4 5 5 2 6 1 7 3 8 6 9 3 6 10 1 2 8 0 0 0 0 4 4 5 5 2 6 1 7 3 8 6 9 3 6 11 1 1 8 0 0 0 0 4 4 5 5 2 6 1 7 3 8 6 9 3 6 11 1 2 8 0 0 0 0 4 4 5 5 2 6 1 7 3 8 6 9 3 6 11 1 1 8 n = 14, 2 solutions, 1X1 squares: 70 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 1 3 0 4 2 5 6 1 7 0 8 2 9 6 5 2 9 10 12 10 10 2 0 0 1 3 0 4 2 5 6 1 7 0 8 2 9 6 5 2 9 10 12 10 13 13 n = 15, 1 solution, 1X1 squares: 83 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 4 4 1 5 3 6 2 7 8 3 9 2 10 2 11 1 12 6 10 9 12 11