Randall's 3x3 semi-magic square of squares
Randall found out many 3x3 semi-magic squares of squares
during his research on
the magic square of squares.
common difference = 3,360 Semi-Magic Square
2 58 82 gcd 2 2 94 113
46 74 94 gcd 2 127 58 46 (1)
97 113 127 gcd 1 74 97 82
Magic Sum = 147^2 Odd diagonal sum = 10,092
common difference = 43,680 Semi-Magic Square
62 218 302 gcd 2 62 313 394
103 233 313 gcd 1 446 218 103 (2)
334 394 446 gcd 2 233 334 302
Magic Sum = 507^2 Odd diagonal sum = 142,572
common difference = 127,680 Semi-Magic Square
146 386 526 gcd 2 146 713 802
503 617 713 gcd 1 878 386 503 (3)
718 802 878 gcd 2 617 718 526
Magic Sum = 1083^2 Odd diagonal sum = 446,988
common difference = 665,280 Semi-Magic Square
102 822 1158 gcd 6 102 1173 3026
213 843 1173 gcd 3 3134 822 213 (4)
2914 3026 3134 gcd 2 843 2914 1158
Magic Sum = 3247^2 Odd diagonal sum = 2,027,052
common difference = 1,145,760 Semi-Magic Square
158 1082 1522 gcd 2 158 1873 2186
1103 1537 1873 gcd 1 2434 1082 1103 (5)
1906 2186 2434 gcd 2 1537 1906 1522
Magic Sum = 2883^2 Odd diagonal sum = 3,512,172
common difference = 1,367,520 Semi-Magic Square
802 1418 1838 gcd 2 802 2722 2969
2162 2458 2722 gcd 2 3191 1418 2162 (6)
2729 2969 3191 gcd 1 2458 2729 1838
Magic Sum = 4107^2 Odd diagonal sum = 6,032,172
common difference = 1,367,520 Semi-Magic Square
802 1418 1838 gcd 2 802 2722 6161
2162 2458 2722 gcd 2 6271 1418 2162 (7)
6049 6161 6271 gcd 1 2458 6049 1838
Magic Sum = 46,010,409 Odd diagonal sum = 6,032,172
common difference = 1,367,520 Semi-Magic Square
802 1418 1838 gcd 2 802 3191 6161
2729 2969 3191 gcd 1 6271 1418 2729 (8)
6049 6161 6271 gcd 1 2969 6049 1838
Magic Sum = 48,783,606 Odd diagonal sum = 6,032,172
common difference = 1,367,520 Semi-Magic Square
2162 2458 2722 gcd 2 2162 3191 6161
2729 2969 3191 gcd 1 6271 2458 2729 (9)
6049 6161 6271 gcd 1 2969 6049 2722
Magic Sum = 52,814,646 Odd diagonal sum = 18,125,292
NOTE: There are 4 triads here, so quartets do exist!
common difference = 2,328,480 Semi-Magic Square
147 1533 2163 gcd 21 147 2866 2562
1886 2426 2866 gcd 2 2982 1533 1886 (10)
2058 2562 2982 gcd 42 2426 2058 2163
Magic Sum = 3847^2 Odd diagonal sum = 7,050,267
common difference = 3,756,480 Semi-Magic Square
562 2018 2798 gcd 2 562 3634 4153
2386 3074 3634 gcd 2 4583 2018 2386 (11)
3673 4153 4583 gcd 1 3074 3673 2798
Magic Sum = 5547^2 Odd diagonal sum = 12,216,972
common difference = 4,514,400 Semi-Magic Square
35 2125 3005 gcd 5 35 3495 2958
1785 2775 3495 gcd 15 3642 2125 1785 (12)
2058 2958 3642 gcd 6 2775 2058 3005
Magic Sum = 20,966,014 Odd diagonal sum = 13,546,875
common difference = 6,726,720 Semi-Magic Square
577 2657 3713 gcd 1 577 4702 5426
2942 3922 4702 gcd 2 6014 2657 2942 (13)
4766 5426 6014 gcd 2 3922 4766 3713
Magic Sum = 7203^2 Odd diagonal sum = 21,178,947
common difference = 7,862,400 Semi-Magic Square
1581 3219 4269 gcd 3 1581 5820 7300
4260 5100 5820 gcd 60 7820 3219 4260 (14)
6740 7300 7820 gcd 20 5100 6740 4269
Magic Sum = 9469^2 Odd diagonal sum = 31,085,883
common difference = 8,168,160 Semi-Magic Square
1426 3194 4286 gcd 2 1426 5081 7753
3079 4201 5081 gcd 1 8263 3194 3079 (15)
7207 7753 8263 gcd 1 4201 7207 4286
Magic Sum = 87,959,046 Odd diagonal sum = 30,604,908
common difference = 8,848,224 Semi-Magic Square
49 2975 4207 gcd 7 49 4318 3885
974 3130 4318 gcd 2 4893 2975 974 (16)
2499 3885 4893 gcd 21 3130 2499 4207
Magic Sum = 33,740,750 Odd diagonal sum = 26,551,875
In light of the 16 Semi-Magic squares found above, it becomes
interesting to conjecture that perhaps a Magic Square of all
squares might exist. After all, there are an infinite number
of Semi-Magic squares. Perhaps one of those is Magic!
Further computer searching for more semi-magic squares is in
progress. Perhaps a parallel Web computer search might be
worthwhile?