The following is a list of all the Super-Symmetric 4x4 Magic Squares. ===================================================================== Magic # calculated as follows: Sum of numbers 1..16 = (1+16)*16/2 = 136 Sum/SideLength = 136/4 = (34) = Magic Number for 4x4 square using 1..16 Unique ways the numbers 1..16 can be summed to 34 using 4 #'s =========================== (01): 01 + 02 + 15 + 16 = 34 (02): 01 + 03 + 14 + 16 = 34 (03): 01 + 04 + 13 + 16 = 34 (04): 01 + 04 + 14 + 15 = 34 (05): 01 + 05 + 12 + 16 = 34 (06): 01 + 05 + 13 + 15 = 34 (07): 01 + 06 + 11 + 16 = 34 (08): 01 + 06 + 12 + 15 = 34 (09): 01 + 06 + 13 + 14 = 34 (10): 01 + 07 + 10 + 16 = 34 (11): 01 + 07 + 11 + 15 = 34 (12): 01 + 07 + 12 + 14 = 34 (13): 01 + 08 + 09 + 16 = 34 (14): 01 + 08 + 10 + 15 = 34 (15): 01 + 08 + 11 + 14 = 34 (16): 01 + 08 + 12 + 13 = 34 (17): 01 + 09 + 10 + 14 = 34 (18): 01 + 09 + 11 + 13 = 34 (19): 01 + 10 + 11 + 12 = 34 (20): 02 + 03 + 13 + 16 = 34 (21): 02 + 03 + 14 + 15 = 34 (22): 02 + 04 + 12 + 16 = 34 (23): 02 + 04 + 13 + 15 = 34 (24): 02 + 05 + 11 + 16 = 34 (25): 02 + 05 + 12 + 15 = 34 (26): 02 + 05 + 13 + 14 = 34 (27): 02 + 06 + 10 + 16 = 34 (28): 02 + 06 + 11 + 15 = 34 (29): 02 + 06 + 12 + 14 = 34 (30): 02 + 07 + 09 + 16 = 34 (31): 02 + 07 + 10 + 15 = 34 (32): 02 + 07 + 11 + 14 = 34 (33): 02 + 07 + 12 + 13 = 34 (34): 02 + 08 + 09 + 15 = 34 (35): 02 + 08 + 10 + 14 = 34 (36): 02 + 08 + 11 + 13 = 34 (37): 02 + 09 + 10 + 13 = 34 (38): 02 + 09 + 11 + 12 = 34 (39): 03 + 04 + 11 + 16 = 34 (40): 03 + 04 + 12 + 15 = 34 (41): 03 + 04 + 13 + 14 = 34 (42): 03 + 05 + 10 + 16 = 34 (43): 03 + 05 + 11 + 15 = 34 (44): 03 + 05 + 12 + 14 = 34 (45): 03 + 06 + 09 + 16 = 34 (46): 03 + 06 + 10 + 15 = 34 (47): 03 + 06 + 11 + 14 = 34 (48): 03 + 06 + 12 + 13 = 34 (49): 03 + 07 + 08 + 16 = 34 (50): 03 + 07 + 09 + 15 = 34 (51): 03 + 07 + 10 + 14 = 34 (52): 03 + 07 + 11 + 13 = 34 (53): 03 + 08 + 09 + 14 = 34 (54): 03 + 08 + 10 + 13 = 34 (55): 03 + 08 + 11 + 12 = 34 (56): 03 + 09 + 10 + 12 = 34 (57): 04 + 05 + 09 + 16 = 34 (58): 04 + 05 + 10 + 15 = 34 (59): 04 + 05 + 11 + 14 = 34 (60): 04 + 05 + 12 + 13 = 34 (61): 04 + 06 + 08 + 16 = 34 (62): 04 + 06 + 09 + 15 = 34 (63): 04 + 06 + 10 + 14 = 34 (64): 04 + 06 + 11 + 13 = 34 (65): 04 + 07 + 08 + 15 = 34 (66): 04 + 07 + 09 + 14 = 34 (67): 04 + 07 + 10 + 13 = 34 (68): 04 + 07 + 11 + 12 = 34 (69): 04 + 08 + 09 + 13 = 34 (70): 04 + 08 + 10 + 12 = 34 (71): 04 + 09 + 10 + 11 = 34 (72): 05 + 06 + 07 + 16 = 34 (73): 05 + 06 + 08 + 15 = 34 (74): 05 + 06 + 09 + 14 = 34 (75): 05 + 06 + 10 + 13 = 34 (76): 05 + 06 + 11 + 12 = 34 (77): 05 + 07 + 08 + 14 = 34 (78): 05 + 07 + 09 + 13 = 34 (79): 05 + 07 + 10 + 12 = 34 (80): 05 + 08 + 09 + 12 = 34 (81): 05 + 08 + 10 + 11 = 34 (82): 06 + 07 + 08 + 13 = 34 (83): 06 + 07 + 09 + 12 = 34 (84): 06 + 07 + 10 + 11 = 34 (85): 06 + 08 + 09 + 11 = 34 (86): 07 + 08 + 09 + 10 = 34 =========================== Every row, every column, both diagnols add to the same number (34), as a standard magic square. However, these squares are super magic. In addition, every 2x2 subsquare adds to 34 The corners of 3x3 add to 34 The corners of 4x4 add to 34 The "extended" diagnols add to 34 There are 384 such 4x4 squares, however, the square has rotation and reflection symmetry, so there are really only 48 unique solutions... The chances of finding one of these at random is: 384/16! = 384/20922789888000 = 1/54486432000 = 1.835319295636756e-11 or about (1 in 50 billion) The solution space was reduced using simple algebra from (16!) to (16 choose 4), then solutions found by iterated yes/no trials. =============================================================================== SuperMagic:1 14 1 12 7 4 15 6 9 5 10 3 16 11 8 13 2 SuperMagic:2 15 1 12 6 4 14 7 9 5 11 2 16 10 8 13 3 SuperMagic:3 14 1 8 11 4 15 10 5 9 6 3 16 7 12 13 2 SuperMagic:4 15 1 8 10 4 14 11 5 9 7 2 16 6 12 13 3 SuperMagic:5 12 1 14 7 6 15 4 9 3 10 5 16 13 8 11 2 SuperMagic:6 15 1 14 4 6 12 7 9 3 13 2 16 10 8 11 5 SuperMagic:7 12 1 8 13 6 15 10 3 9 4 5 16 7 14 11 2 SuperMagic:8 15 1 8 10 6 12 13 3 9 7 2 16 4 14 11 5 SuperMagic:9 12 1 15 6 7 14 4 9 2 11 5 16 13 8 10 3 SuperMagic:10 14 1 15 4 7 12 6 9 2 13 3 16 11 8 10 5 SuperMagic:11 12 1 8 13 7 14 11 2 9 4 5 16 6 15 10 3 SuperMagic:12 14 1 8 11 7 12 13 2 9 6 3 16 4 15 10 5 SuperMagic:13 8 1 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