Minimal defect = 2n-# for a grid size n and a symmetry-class n iden rot2 dia1 ort1 rot4 dia2 ort2 full n -------------------------------------------------------------------------------- 1 - - - - - - - 1(1) 1 2 - - 1(2) 2(1) - 2(1) - 0(1)[4(1)] 2 3 1(2) 2(1) 1(2) 1(1) - 0(1)[2(0)] 4(1) 2(2) 3 4 1(28) 0 1(5) 0(1)[2(9)] 4(1) 0(1)[2(3)] 4(1) 0(1)[4(2)] 4 5 0 2(15) 0 1(2) 2(1) 2(4) 2(1) 2(2) 5 6 0 0 1(18) 2(42) 0 0(2)[2(9)] 4(6) 4(3) 6 7 0 0 0 1(6) 2 2(11) 2(6) 2(2) 7 8 0 0 0 0(1)[2(86)] 0 2(10) 4(11) 4(6) 8 9 0 0 0 1(10) 2 2(17) 2(5) 2(3) 9 10 0 0 0 2(246) 0 0(1)[2(21)] 4(33) 0(1)[4(2)] 10 11 0 0 0 1(8) 2 2(38) 2(5) 2(3) 11 12 0 0 0 2(457) 0 0(2)[2(31)] 4(35) 4(10) 12 13 0 0 0 1(35) 2 0(1)[2(72)] 2(12) 2(4) 13 14 0 0 0 2(381) 0 0(3)[2(26)] 4(17) 4(3) 14 15 0 0 0 1(37) 2 0(1)[2(90)] 2(10) 2(2) 15 16 0 0 0 2(1487) 0 0(1)[2(36)] 4(49) 4(5) 16 17 0 0 0 1(25) 2 2(76) 2(7) 2(3) 17 18 0 0 0 2(1522) 0 0(2)[2(46)] 4(39) 4(1) 18 19 0 0 0 1(37) 2 2(115) 2(3) 6(50) 19 20 0 0 2(2345) 0 0(2)[2(66)] 4(42) 8(19) 20 21 0 0 1(92) 2 2(140) 2(10) 6(31) 21 22 0 0 2(3624) 0 0(1)[2(81)] 4(47) 4(1) 22 23 0 0 1(45) 2 0(1)[2(189)] 2(5) 6(33) 23 24 0 0 2(4713) 0 0(2)[2(86)] 4(27) 4(1) 24 25 0 0 1(80) 2 0(2)[2(226)] 2(7) 2(1) 25 26 0 0 0(1)2(5209) 0 2(76) 4(50) 8(70) 26 27 0 0 1(60) 2 2(220) 6(29) 27 28 0(1) 0 2(70) 4(41) 4(1) 28 29 0 2 0(1)[2(276)] 6(10) 29 30 2 0 0(1)[2(77)] 4(30) 8(55) 30 31 0 2 0(2)[2(308)] 6(25) 31 32 2 0 0(1)[2(80)] 4(8) 8(6) 32 33 0 2 2(362) 2(1) 33 34 >= 2 0 2(75) 4(10) 8(35) 34 35 0 2 0(1)[2(330)] 6 35 36 0 0(1)[2(102)] 4(26) 8(7) 36 37 0 2 2(358) 6 37 38 0 0(1)[2(86)] 4(14) 8(20) 38 39 0 2 2(342) 6 39 40 0 2(78) 4(13) 8(8) 40 41 0 2 2(370) 41 42 0 0(1) 4(6) 8(19) 42 43 0 2 2 43 44 0 0(1) 4(5) 8(4) 44 45 0 2 2 45 46 0 2 4(9) 8(12) 46 47 0 2 0(1) 47 48 0 4 8(3) 48 49 0 2 49 50 0 >= 4 8(8) 50 51 0 2 51 52 0 >= 4 8(2) 52 53 0 2 53 54 0 >= 4 8(4) 54 55 0 2 55 56 0 8(1) 56 57 0 2 57 58 0 8(4) 58 59 0 2 59 60 0 8(1) 60 61 0 2 61 62 0 8(2) 62 63 0 2 63 64 0 8(1) 64 65 65 66 0 8(1) 66 67 67 68 0 12(186) 68 69 69 70 8(2) 70 71 71 72 12(123) 72 73 73 74 8(2) 74 75 75 76 12 76 77 77 78 12 78 79 79 80 12 80 81 81 82 82 83 83 84 84 85 85 86 86 87 87 88 88 89 89 90 90 91 91 92 92 93 93 94 94 95 95 96 96 97 97 98 98 Column headings are the gridsize n and the 8 different symmetry-classes of the square, namely iden, rot2, dia1, ort1, rot4, dia2, ort2 and full. The numbers in the table-entries are the minimal defect=2*n-# such that #-many number of points can be chosen such that the no-three-in-line condition is satisfied for these specific points in the n times n Cartesian grid and the given symmetry-class. Thus only if the defect equals 0 there is a solution for the no-three-in-line problem! A known lower bound is given as ">= at_least_value". In parenthesis () is given the number of different solutions for the minimal defect; in brackets [] is given the counts for next greater defect. last update: 2026-05-03