Magic Square of Cubes Solutions
4x4 Magic Square of Cubes of Distinct Integers
4x4 Semi-Magic Squares of Cubes of Distinct Positive Integers
5x5 Magic Square of Distinct Cubes
5x5 Semi-Magic Squares of Distinct Positive Cubes
6x6 Magic Square of Distinct Cubes
7x7 Magic Square of Distinct Cubes
8x8 Semi-Magic Square of Distinct Positive Cubes
4x4 Magic Square of Cubes of Distinct Integers
All of the known 4x4 magic squares of cubes use the following structure
or its transpose.
A3 B3 C3 D3
E3 F3 G3 H3
-E3 -F3 -G3 -H3
-A3 -B3 -C3 -D3
This is the smallest solution.
193 -33 -103 -183
-423 213 283 353
423 -213 -283 -353
-193 33 103 183
It is an open problem whether there exists a 4x4 magic square of cubes
of distinct positive integers. It is also an open problem whether there
is a solution with any distinct integers not having the above structure.
There is a related problem which provably has this same kind of solution set.
It is three cubes in arithmetic progression: x3 + y3 = 2z3.
There is no solution in distinct positive integers
and the only solutions allowing negative integers have z = 0 and x = -y.
Can the method of proof used by the three cubes problem be used to prove
that the 4x4 magic square of cubes problem must have the above structure?
4x4 Semi-Magic Squares of Cubes of Distinct Positive Integers
Seven of the eight smallest solutions use the following structure.
(af)3 (de)3 (ce)3 (bf)3,
(bh)3 (cg)3 (dg)3 (ah)3,
(bg)3 (ch)3 (dh)3 (ag)3,
(ae)3 (df)3 (cf)3 (be)3,
where
a3 + b3 = c3 + d3 = p,
e3 + f3 = g3 + h3 = q.
with magic sum = pq.
(MaxNb = 192, S3 = 7,095,816 = 1729 x 4104)
243 1443 23 1603
1923 183 163 203
153 903 1803 813
93 1503 1083 1353
(MaxNb = 313, S3 = 42,699,384)
3133 953 1273 2093
1353 2973 733 2393
893 1093 2753 2713
2073 2433 2693 253
(MaxNb = 324, S3 = 35,760,907 = 1729 x 20,683)
3243 273 1003 903
243 2883 1713 1903
1203 103 2703 2433
193 2283 2163 2403
(MaxNb = 408, S3 = 67,970,448 = 1729 x 39,312)
4083 333 243 153
343 3963 23 1803
203 1353 3403 2973
183 1503 3063 3303
(MaxNb = 408, S3 = 69,217,057 = 1729 x 40,033)
4083 343 903 813
333 3963 1443 1603
1083 93 3403 3063
163 1923 2973 3303
(MaxNb = 432, S3 = 84,883,032 = 4104 x 20,683)
4323 1603 483 383
1503 4053 1713 2163
543 203 3843 3043
903 2433 2853 3603
(MaxNb = 468, S3 = 111,057,128 = 1729 x 64,232)
4683 393 1703 1533
363 4323 2343 2603
2043 173 3903 3513
263 3123 3243 3603
(MaxNb = 480, S3 = 113,643,712 = 1729 x 65,728)
4803 1443 333 313
1203 4003 2793 2973
403 123 3963 3723
1083 3603 3103 3303
5x5 Magic Square of Distinct Cubes
The smallest solution has an associated structure.
113 -203 123 133 143
-153 213 33 -103 -173
-53 -43 03 43 53
173 103 -33 -213 153
-143 -133 -123 203 -113
5x5 Semi-Magic Squares of Distinct Positive Cubes
(MaxNb=110, S3=1,408,896)
1103 103 403 233 93
33 1013 183 483 643
213 673 983 63 543
363 143 703 973 473
283 423 383 723 963
(MaxNb=141, S3=4,416,616)
1413 753 823 163 863
393 1293 43 903 1143
253 713 1283 1243 343
263 583 1183 1163 1003
1153 1093 503 603 1083
6x6 Magic Square of Distinct Cubes
183 33 -133 133 -33 -183
13 -143 -83 -13 143 83
-103 173 -43 43 -173 103
-163 -53 153 -153 53 163
-93 -73 93 -63 73 63
-23 -123 -113 -113 123 23
7x7 Magic Square of Distinct Cubes
-173 -63 -213 203 123 193 -133
93 73 -153 -13 103 113 -33
-183 -163 83 -53 43 63 213
243 233 223 03 -243 -233 -223
-143 -193 133 53 173 -123 163
-103 -113 -93 13 153 -73 33
-43 -23 23 -203 143 183 -83
8x8 Semi-Magic Square of Distinct Positive Cubes
using distinct positive integers 1...653.
(1...643 is not possible).
(magic sum = 551,979)
653 643 93 123 63 233 73 33
23 293 633 253 453 133 513 333
203 483 193 623 173 153 523 343
283 223 383 323 613 373 413 443
103 113 603 403 273 423 303 533
433 213 263 463 243 593 163 503
83 313 183 353 583 543 143 493
553 473 43 393 53 363 563 13