AG(4,3): n points, the best layout found (a hyperplane, plus points in
the two parallel ones) against the most the counting argument allows
n= 3  built   1  proved at most   1  TIGHT    (no split allowed; splits allowed: [])
n= 4  built   1  proved at most   1  TIGHT    (no split allowed; splits allowed: [])
n= 5  built   2  proved at most   2  TIGHT    (no split allowed; splits allowed: [])
n= 6  built   3  proved at most   3  TIGHT    (pairs alone (too many); splits allowed: [(2, 2, 2)])
n= 7  built   5  proved at most   5  TIGHT    (no split allowed; splits allowed: [])
n= 8  built   8  proved at most   8  TIGHT    (no split allowed; splits allowed: [])
n= 9  built  12  proved at most  12  TIGHT    (no split allowed; splits allowed: [])
n=10  built  12  proved at most  12  TIGHT    (no split allowed; splits allowed: [])
n=11  built  13  proved at most  13  TIGHT    (pairs alone (too many); splits allowed: [(4, 4, 3)])
n=12  built  14  proved at most  14  TIGHT    (pairs alone (too many); splits allowed: [(4, 4, 4), (5, 4, 3)])
n=13  built  16  proved at most  16  TIGHT    (pairs alone (too many); splits allowed: [(5, 4, 4), (5, 5, 3), (6, 4, 3)])
n=14  built  19  proved at most  19  TIGHT    (pairs alone (too many); splits allowed: [(5, 5, 4), (6, 4, 4), (6, 5, 3)])
n=15  built  23  proved at most  23  TIGHT    (pairs alone (too many); splits allowed: [(5, 5, 5), (6, 5, 4), (6, 6, 3)])
n=16  built  26  proved at most  26  TIGHT    (triples alone; splits allowed: [(6, 5, 5), (6, 6, 4), (7, 5, 4), (7, 6, 3)])
n=17  built  30  proved at most  30  TIGHT    (triples alone; splits allowed: [(6, 6, 5), (7, 5, 5), (7, 6, 4), (7, 7, 3), (8, 5, 4)])
n=18  built  36  proved at most  36  TIGHT    (pairs alone (too many); splits allowed: [(6, 6, 6), (7, 6, 5), (7, 7, 4), (8, 5, 5)])
n=19  built  41  proved at most  41  TIGHT    (pairs alone (too many); splits allowed: [(7, 6, 6), (7, 7, 5), (8, 6, 5), (8, 7, 4)])
n=20  built  47  proved at most  47  TIGHT    (triples alone; splits allowed: [(7, 7, 6), (8, 6, 6), (8, 7, 5), (8, 8, 4)])
n=21  built  54  proved at most  54  TIGHT    (pairs alone (too many); splits allowed: [(7, 7, 7), (8, 7, 6), (8, 8, 5)])
n=22  built  62  proved at most  62  TIGHT    (pairs alone (too many); splits allowed: [(8, 7, 7), (8, 8, 6)])
n=23  built  71  proved at most  71  TIGHT    (pairs alone (too many); splits allowed: [(8, 8, 7)])
n=24  built  81  proved at most  81  TIGHT    (pairs alone (too many); splits allowed: [(8, 8, 8)])
n=25  built  92  proved at most  92  TIGHT    (no split allowed; splits allowed: [])
n=26  built 104  proved at most 104  TIGHT    (no split allowed; splits allowed: [])
n=27  built 117  proved at most 117  TIGHT    (no split allowed; splits allowed: [])
n=28  built 117  proved at most 117  TIGHT    (no split allowed; splits allowed: [])
n=29  built 118  proved at most 118  TIGHT    (pairs alone (too many); splits allowed: [(10, 10, 9)])
n=30  built 119  proved at most 119  TIGHT    (pairs alone (too many); splits allowed: [(10, 10, 10), (11, 10, 9), (11, 11, 8)])
n=31  built 121  proved at most 121  TIGHT    (pairs alone (too many); splits allowed: [(11, 10, 10), (11, 11, 9), (12, 10, 9), (12, 11, 8)])
n=32  built 124  proved at most 124  TIGHT    (pairs alone (too many); splits allowed: [(11, 11, 10), (12, 10, 10), (12, 11, 9), (12, 12, 8), (13, 10, 9), (13, 11, 8)])
n=33  built 128  proved at most 128  TIGHT    (triples alone; splits allowed: [(11, 11, 11), (12, 11, 10), (12, 12, 9), (13, 10, 10), (13, 11, 9), (13, 12, 8), (14, 10, 9)])
n=34  built 131  proved at most 131  TIGHT    (both together; splits allowed: [(12, 11, 11), (12, 12, 10), (13, 11, 10), (13, 12, 9), (13, 13, 8), (14, 10, 10), (14, 11, 9), (14, 12, 8), (15, 10, 9), (15, 11, 8)])
n=35  built 135  proved at most 135  TIGHT    (both together; splits allowed: [(12, 12, 11), (13, 11, 11), (13, 12, 10), (13, 13, 9), (14, 11, 10), (14, 12, 9), (14, 13, 8), (14, 14, 7), (15, 10, 10), (15, 11, 9), (15, 12, 8), (16, 10, 9)])
n=36  built 140  proved at most 140  TIGHT    (both together; splits allowed: [(12, 12, 12), (13, 12, 11), (13, 13, 10), (14, 11, 11), (14, 12, 10), (14, 13, 9), (14, 14, 8), (15, 11, 10), (15, 12, 9), (15, 13, 8), (15, 14, 7), (16, 10, 10), (16, 11, 9), (16, 12, 8), (17, 10, 9), (18, 9, 9)])
n=37  built 146  proved at most 146  TIGHT    (both together; splits allowed: [(13, 12, 12), (13, 13, 11), (14, 12, 11), (14, 13, 10), (14, 14, 9), (15, 11, 11), (15, 12, 10), (15, 13, 9), (15, 14, 8), (15, 15, 7), (16, 11, 10), (16, 12, 9), (16, 13, 8), (16, 14, 7), (17, 10, 10), (17, 11, 9), (17, 12, 8), (18, 10, 9)])
n=38  built 152  proved at most 152  TIGHT    (both together; splits allowed: [(13, 13, 12), (14, 12, 12), (14, 13, 11), (14, 14, 10), (15, 12, 11), (15, 13, 10), (15, 14, 9), (15, 15, 8), (16, 11, 11), (16, 12, 10), (16, 13, 9), (16, 14, 8), (16, 15, 7), (17, 11, 10), (17, 12, 9), (17, 13, 8), (18, 10, 10), (18, 11, 9), (18, 12, 8), (19, 10, 9)])
n=39  built 159  proved at most 159  TIGHT    (both together; splits allowed: [(13, 13, 13), (14, 13, 12), (14, 14, 11), (15, 12, 12), (15, 13, 11), (15, 14, 10), (15, 15, 9), (16, 12, 11), (16, 13, 10), (16, 14, 9), (16, 15, 8), (16, 16, 7), (17, 11, 11), (17, 12, 10), (17, 13, 9), (17, 14, 8), (17, 15, 7), (18, 11, 10), (18, 12, 9), (18, 13, 8), (19, 10, 10), (19, 11, 9), (20, 10, 9)])
n=40  built 167  proved at most 167  TIGHT    (both together; splits allowed: [(14, 13, 13), (14, 14, 12), (15, 13, 12), (15, 14, 11), (15, 15, 10), (16, 12, 12), (16, 13, 11), (16, 14, 10), (16, 15, 9), (16, 16, 8), (17, 12, 11), (17, 13, 10), (17, 14, 9), (17, 15, 8), (17, 16, 7), (18, 11, 11), (18, 12, 10), (18, 13, 9), (18, 14, 8), (18, 15, 7), (19, 11, 10), (19, 12, 9), (19, 13, 8), (20, 10, 10), (20, 11, 9), (21, 10, 9)])
n=41  built 176  proved at most 176  TIGHT    (both together; splits allowed: [(14, 14, 13), (15, 13, 13), (15, 14, 12), (15, 15, 11), (16, 13, 12), (16, 14, 11), (16, 15, 10), (16, 16, 9), (17, 12, 12), (17, 13, 11), (17, 14, 10), (17, 15, 9), (17, 16, 8), (17, 17, 7), (18, 12, 11), (18, 13, 10), (18, 14, 9), (18, 15, 8), (18, 16, 7), (19, 11, 11), (19, 12, 10), (19, 13, 9), (19, 14, 8), (20, 11, 10), (20, 12, 9), (21, 10, 10), (21, 11, 9)])
n=42  built 186  proved at most 186  TIGHT    (both together; splits allowed: [(14, 14, 14), (15, 14, 13), (15, 15, 12), (16, 13, 13), (16, 14, 12), (16, 15, 11), (16, 16, 10), (17, 13, 12), (17, 14, 11), (17, 15, 10), (17, 16, 9), (17, 17, 8), (18, 12, 12), (18, 13, 11), (18, 14, 10), (18, 15, 9), (18, 16, 8), (18, 17, 7), (19, 12, 11), (19, 13, 10), (19, 14, 9), (19, 15, 8), (20, 11, 11), (20, 12, 10), (20, 13, 9), (21, 11, 10), (21, 12, 9)])
n=43  built 197  proved at most 197  TIGHT    (both together; splits allowed: [(15, 14, 14), (15, 15, 13), (16, 14, 13), (16, 15, 12), (16, 16, 11), (17, 13, 13), (17, 14, 12), (17, 15, 11), (17, 16, 10), (17, 17, 9), (18, 13, 12), (18, 14, 11), (18, 15, 10), (18, 16, 9), (18, 17, 8), (18, 18, 7), (19, 12, 12), (19, 13, 11), (19, 14, 10), (19, 15, 9), (19, 16, 8), (20, 12, 11), (20, 13, 10), (20, 14, 9), (20, 15, 8), (21, 11, 11), (21, 12, 10), (21, 13, 9)])
n=44  built 209  proved at most 209  TIGHT    (both together; splits allowed: [(15, 15, 14), (16, 14, 14), (16, 15, 13), (16, 16, 12), (17, 14, 13), (17, 15, 12), (17, 16, 11), (17, 17, 10), (18, 13, 13), (18, 14, 12), (18, 15, 11), (18, 16, 10), (18, 17, 9), (18, 18, 8), (19, 13, 12), (19, 14, 11), (19, 15, 10), (19, 16, 9), (19, 17, 8), (20, 12, 12), (20, 13, 11), (20, 14, 10), (20, 15, 9), (21, 12, 11), (21, 13, 10), (21, 14, 9)])
n=45  built 222  proved at most 222  TIGHT    (both together; splits allowed: [(15, 15, 15), (16, 15, 14), (16, 16, 13), (17, 14, 14), (17, 15, 13), (17, 16, 12), (17, 17, 11), (18, 14, 13), (18, 15, 12), (18, 16, 11), (18, 17, 10), (18, 18, 9), (19, 13, 13), (19, 14, 12), (19, 15, 11), (19, 16, 10), (19, 17, 9), (19, 18, 8), (20, 13, 12), (20, 14, 11), (20, 15, 10), (20, 16, 9), (21, 12, 12), (21, 13, 11), (21, 14, 10), (21, 15, 9)])
n=46  built 231  proved at most 231  TIGHT    (both together; splits allowed: [(16, 15, 15), (16, 16, 14), (17, 15, 14), (17, 16, 13), (17, 17, 12), (18, 14, 14), (18, 15, 13), (18, 16, 12), (18, 17, 11), (18, 18, 10), (19, 14, 13), (19, 15, 12), (19, 16, 11), (19, 17, 10), (19, 18, 9), (19, 19, 8), (20, 13, 13), (20, 14, 12), (20, 15, 11), (20, 16, 10), (20, 17, 9), (20, 18, 8), (21, 13, 12), (21, 14, 11), (21, 15, 10), (21, 16, 9), (22, 12, 12), (22, 13, 11), (22, 14, 10), (22, 15, 9)])
n=47  built 241  proved at most 241  TIGHT    (both together; splits allowed: [(16, 16, 15), (17, 15, 15), (17, 16, 14), (17, 17, 13), (18, 15, 14), (18, 16, 13), (18, 17, 12), (18, 18, 11), (19, 14, 14), (19, 15, 13), (19, 16, 12), (19, 17, 11), (19, 18, 10), (19, 19, 9), (20, 14, 13), (20, 15, 12), (20, 16, 11), (20, 17, 10), (20, 18, 9), (20, 19, 8), (21, 13, 13), (21, 14, 12), (21, 15, 11), (21, 16, 10), (21, 17, 9), (21, 18, 8), (22, 13, 12), (22, 14, 11), (22, 15, 10), (22, 16, 9), (23, 12, 12), (23, 13, 11), (23, 14, 10)])
n=48  built 252  proved at most 253  gap 1    (both together; splits allowed: [(16, 16, 16), (17, 16, 15), (17, 17, 14), (18, 15, 15), (18, 16, 14), (18, 17, 13), (18, 18, 12), (19, 15, 14), (19, 16, 13), (19, 17, 12), (19, 18, 11), (19, 19, 10), (20, 14, 14), (20, 15, 13), (20, 16, 12), (20, 17, 11), (20, 18, 10), (20, 19, 9), (20, 20, 8), (21, 14, 13), (21, 15, 12), (21, 16, 11), (21, 17, 10), (21, 18, 9), (21, 19, 8), (22, 13, 13), (22, 14, 12), (22, 15, 11), (22, 16, 10), (22, 17, 9), (23, 13, 12), (23, 14, 11), (23, 15, 10)])
n=49  built 264  proved at most 267  gap 3    (both together; splits allowed: [(17, 16, 16), (17, 17, 15), (18, 16, 15), (18, 17, 14), (18, 18, 13), (19, 15, 15), (19, 16, 14), (19, 17, 13), (19, 18, 12), (19, 19, 11), (20, 15, 14), (20, 16, 13), (20, 17, 12), (20, 18, 11), (20, 19, 10), (20, 20, 9), (21, 14, 14), (21, 15, 13), (21, 16, 12), (21, 17, 11), (21, 18, 10), (21, 19, 9), (21, 20, 8), (22, 14, 13), (22, 15, 12), (22, 16, 11), (22, 17, 10), (22, 18, 9), (23, 13, 13), (23, 14, 12), (23, 15, 11), (23, 16, 10)])
n=50  built 276  proved at most 281  gap 5    (both together; splits allowed: [(17, 17, 16), (18, 16, 16), (18, 17, 15), (18, 18, 14), (19, 16, 15), (19, 17, 14), (19, 18, 13), (19, 19, 12), (20, 15, 15), (20, 16, 14), (20, 17, 13), (20, 18, 12), (20, 19, 11), (20, 20, 10), (21, 15, 14), (21, 16, 13), (21, 17, 12), (21, 18, 11), (21, 19, 10), (21, 20, 9), (21, 21, 8), (22, 14, 14), (22, 15, 13), (22, 16, 12), (22, 17, 11), (22, 18, 10), (22, 19, 9), (23, 14, 13), (23, 15, 12), (23, 16, 11), (23, 17, 10), (24, 13, 13), (24, 14, 12), (24, 15, 11)])
n=51  built 289  proved at most 295  gap 6    (both together; splits allowed: [(17, 17, 17), (18, 17, 16), (18, 18, 15), (19, 16, 16), (19, 17, 15), (19, 18, 14), (19, 19, 13), (20, 16, 15), (20, 17, 14), (20, 18, 13), (20, 19, 12), (20, 20, 11), (21, 15, 15), (21, 16, 14), (21, 17, 13), (21, 18, 12), (21, 19, 11), (21, 20, 10), (21, 21, 9), (22, 15, 14), (22, 16, 13), (22, 17, 12), (22, 18, 11), (22, 19, 10), (22, 20, 9), (23, 14, 14), (23, 15, 13), (23, 16, 12), (23, 17, 11), (23, 18, 10), (24, 14, 13), (24, 15, 12), (24, 16, 11)])
n=52  built 303  proved at most 309  gap 6    (both together; splits allowed: [(18, 17, 17), (18, 18, 16), (19, 17, 16), (19, 18, 15), (19, 19, 14), (20, 16, 16), (20, 17, 15), (20, 18, 14), (20, 19, 13), (20, 20, 12), (21, 16, 15), (21, 17, 14), (21, 18, 13), (21, 19, 12), (21, 20, 11), (21, 21, 10), (22, 15, 15), (22, 16, 14), (22, 17, 13), (22, 18, 12), (22, 19, 11), (22, 20, 10), (22, 21, 9), (23, 15, 14), (23, 16, 13), (23, 17, 12), (23, 18, 11), (23, 19, 10), (23, 20, 9), (24, 14, 14), (24, 15, 13), (24, 16, 12), (24, 17, 11)])
n=53  built 318  proved at most 326  gap 8    (both together; splits allowed: [(18, 18, 17), (19, 17, 17), (19, 18, 16), (19, 19, 15), (20, 17, 16), (20, 18, 15), (20, 19, 14), (20, 20, 13), (21, 16, 16), (21, 17, 15), (21, 18, 14), (21, 19, 13), (21, 20, 12), (21, 21, 11), (22, 16, 15), (22, 17, 14), (22, 18, 13), (22, 19, 12), (22, 20, 11), (22, 21, 10), (22, 22, 9), (23, 15, 15), (23, 16, 14), (23, 17, 13), (23, 18, 12), (23, 19, 11), (23, 20, 10), (24, 15, 14), (24, 16, 13), (24, 17, 12), (24, 18, 11)])
n=54  built 334  proved at most 344  gap 10   (both together; splits allowed: [(18, 18, 18), (19, 18, 17), (19, 19, 16), (20, 17, 17), (20, 18, 16), (20, 19, 15), (20, 20, 14), (21, 17, 16), (21, 18, 15), (21, 19, 14), (21, 20, 13), (21, 21, 12), (22, 16, 16), (22, 17, 15), (22, 18, 14), (22, 19, 13), (22, 20, 12), (22, 21, 11), (22, 22, 10), (23, 16, 15), (23, 17, 14), (23, 18, 13), (23, 19, 12), (23, 20, 11), (23, 21, 10), (24, 15, 15), (24, 16, 14), (24, 17, 13), (24, 18, 12)])
n=55  built 351  proved at most 360  gap 9    (both together; splits allowed: [(19, 18, 18), (19, 19, 17), (20, 18, 17), (20, 19, 16), (20, 20, 15), (21, 17, 17), (21, 18, 16), (21, 19, 15), (21, 20, 14), (21, 21, 13), (22, 17, 16), (22, 18, 15), (22, 19, 14), (22, 20, 13), (22, 21, 12), (22, 22, 11), (23, 16, 16), (23, 17, 15), (23, 18, 14), (23, 19, 13), (23, 20, 12), (23, 21, 11), (23, 22, 10), (23, 23, 9), (24, 16, 15), (24, 17, 14), (24, 18, 13), (24, 19, 12), (24, 20, 11), (25, 15, 15), (25, 16, 14)])
n=56  built 369  proved at most 376  gap 7    (both together; splits allowed: [(19, 19, 18), (20, 18, 18), (20, 19, 17), (20, 20, 16), (21, 18, 17), (21, 19, 16), (21, 20, 15), (21, 21, 14), (22, 17, 17), (22, 18, 16), (22, 19, 15), (22, 20, 14), (22, 21, 13), (22, 22, 12), (23, 17, 16), (23, 18, 15), (23, 19, 14), (23, 20, 13), (23, 21, 12), (23, 22, 11), (23, 23, 10), (24, 16, 16), (24, 17, 15), (24, 18, 14), (24, 19, 13), (24, 20, 12), (24, 21, 11), (25, 16, 15), (25, 17, 14), (25, 18, 13)])
n=57  built 388  proved at most 396  gap 8    (both together; splits allowed: [(19, 19, 19), (20, 19, 18), (20, 20, 17), (21, 18, 18), (21, 19, 17), (21, 20, 16), (21, 21, 15), (22, 18, 17), (22, 19, 16), (22, 20, 15), (22, 21, 14), (22, 22, 13), (23, 17, 17), (23, 18, 16), (23, 19, 15), (23, 20, 14), (23, 21, 13), (23, 22, 12), (23, 23, 11), (24, 17, 16), (24, 18, 15), (24, 19, 14), (24, 20, 13), (24, 21, 12), (24, 22, 11), (25, 16, 16), (25, 17, 15), (25, 18, 14)])
n=58  built 406  proved at most 412  gap 6    (both together; splits allowed: [(20, 19, 19), (20, 20, 18), (21, 19, 18), (21, 20, 17), (21, 21, 16), (22, 18, 18), (22, 19, 17), (22, 20, 16), (22, 21, 15), (22, 22, 14), (23, 18, 17), (23, 19, 16), (23, 20, 15), (23, 21, 14), (23, 22, 13), (23, 23, 12), (24, 17, 17), (24, 18, 16), (24, 19, 15), (24, 20, 14), (24, 21, 13), (24, 22, 12), (24, 23, 11), (24, 24, 10), (25, 17, 16), (25, 18, 15), (25, 19, 14), (25, 20, 13)])
n=59  built 425  proved at most 432  gap 7    (both together; splits allowed: [(20, 20, 19), (21, 19, 19), (21, 20, 18), (21, 21, 17), (22, 19, 18), (22, 20, 17), (22, 21, 16), (22, 22, 15), (23, 18, 18), (23, 19, 17), (23, 20, 16), (23, 21, 15), (23, 22, 14), (23, 23, 13), (24, 18, 17), (24, 19, 16), (24, 20, 15), (24, 21, 14), (24, 22, 13), (24, 23, 12), (24, 24, 11), (25, 17, 17), (25, 18, 16), (25, 19, 15), (25, 20, 14), (25, 21, 13), (26, 18, 15)])
n=60  built 446  proved at most 454  gap 8    (both together; splits allowed: [(20, 20, 20), (21, 20, 19), (21, 21, 18), (22, 19, 19), (22, 20, 18), (22, 21, 17), (22, 22, 16), (23, 19, 18), (23, 20, 17), (23, 21, 16), (23, 22, 15), (23, 23, 14), (24, 18, 18), (24, 19, 17), (24, 20, 16), (24, 21, 15), (24, 22, 14), (24, 23, 13), (24, 24, 12), (25, 18, 17), (25, 19, 16), (25, 20, 15), (25, 21, 14), (25, 22, 13)])
