Squares on a Cube

 Fail

About Thomas' mathematical adventures:

I have a passion for recreational mathematical puzzles. Especially combinatorical puzzles or puzzles involving square numbers. My approach is normally some mathematical research combined with brute force CPU power. Often I am not able to solve the puzzle, but sometimes I am. However even when I fail to find a solution the information gained can be interesting since it can reveal the minimum size of a solution or revealing patterns in the numbers. This is the first page in my series of 'mathematic adventures', which unfortunately is a fail! I have currently about 5-6 adventures to post and two of them are solved. Over the next weeks I plan to describe them as well.

Abstract:

Is it possible to write 6 different square numbers on each face of a cube, such that the sum of the squares on the 3 faces around each of the 8 corners are also adding to a square number?
This problem is from recreational mathematics and has yet not been solved. Currently it is the only unsolved square problem on The Contest Center. I was not able to find a solution, but found 39 different partial solutions where the condition for the 8 corner sums only failed to be a square for the last corner. Finding the partial solutions was quite hard also, as they are still rare. I will not probably not make further attemps to solve this puzzle.

Dice

Definitions:

A cube has the following 8 corners: C1=(1,2,3), C2=(1,2,4), C3=(1,3,5), C4=(1,4,5), C5=(2,3,6) ,C6=(2,4,6), C7=(3,5,6) and C8=(4,5,6)
Define the value at face 1 to be S12, where S1 is an integer. And so on for S2,S3,S4, S5 and S6
Define the sum of the 3 faces meeting at C1 to be Sum1=S12+S22+S32 and so on for Sn. That is Sum1=S12+S22+S32, ... ,Sum8=S42+S52+S62
Define by a partial solution a solution where only Sum8 fails to be a square, while Sum1,..., Sum7 are squares.

Results:

This search for a solution was almost a 100% brute force attack. I was not able to find any usefull mathematical results. For the brute force attack you can assume WLOG that S1 is the smallest number.
That is S1 < S2 & S1 < S3 & S1 < S4 & S1 < S5 & S1 < S6
And you can also assume WLOG that S1 < S2 < S3 < S4. (note this does not involve S5 and S6)
The brute force attack was done for the following ranges:

Algorithm:

The number of combinations for S1,S2,...,S6 I have covered in my search is ~10e30. This is an astronomical number and is not feasible by stupid brute force of course. But I did not have to search all combinations since many can be exluded by logic. I start by finding (S1,S2,S3) such that Sum1 is a square. If Sum1 is not a square, there is no reason to search further for S4,S5,S6. For these (S1,S2,S3)-triplets, I search for S4 values such that Sum2 is also a square, which only is fullfilled for a small set of S4's. Next I search for S5 such that satisfy that both Sum3 and S4 is a square. This greatly reduces number of valid S5 values and rarely have any solutions at all. And next is only test all possible values S6's such that the rest of the Sumn equations holds.

The smallest partial solution I found is the following(example highlighted in the table below):

S1=63
S2=294
S3=378
S4=686
S5=54
S6=1323

Sum1=632+2942+3782=4832
Sum2=632+2942+6862=7492
Sum3=632+3782+542=3872
Sum4=632+6862542=6912
Sum5=2942+3782+13232=14072
Sum6=2942+6862+13232=15192
Sum7=3782+542+13232=13772
Sum8=6862+542+13232=2223841 ← NOT SQUARE 		Dice

List of the 39 partial solutions found by brute force.

The list only shows partial solutions that are not k2-transformation of another partial solution. I have also listed the non-square prime-factors of Sum8.
S1 S2 S3 S4 S5 S6 Sum1 Sum2 Sum3 Sum4 Sum5 Sum6 Sum7 Sum8 Non-square prime-factors in Sum8
07562592101923494450589 (2700)2(10220)2(35040)2(36400)2(50661)2(51611)2(61539)23884206921 79269529
0123232764224112322145 (3500)2(4400)2(11700)2(12000)2(4105)2(4895)2(11895)2148601025 660449
050405775528006050055692 (7665)2(53040)2(60775)2(80300)2(56217)2(76908)2(82433)29549688864 2, 29, 10290613
07140110881589524684147600 (13188)2(17425)2(27060)2(29359)2(148188)2(148625)2(150060)222647710881 1381, 5309, 3089
08160117812240032340118008 (14331)2(23840)2(34419)2(39340)2(118875)2(120392)2(122925)215473523664 1493, 647753
0110401593912672018295237100 (19389)2(127200)2(183645)2(222552)2(41861)2(132500)2(187355)250905802704 41, 77600309
01320013860362880381024416075 (19140)2(363120)2(381276)2(526176)2(416515)2(552245)2(564349)2449979588601 53, 51949, 163433
013728201963650453703149600 (24420)2(39000)2(57375)2(64935)2(151580)2(154600)2(160225)226596714225 1063868569
01713618700291060317625277680 (25364)2(291564)2(318175)2(430815)2(278836)2(402636)2(422305)2262707746625 5, 1913, 122069
03197748840270864175750156600 (58377)2(272745)2(182410)2(322886)2(167127)2(314505)2(240410)2128778928996 38281489
0480487536131046448694881900 (89375)2(314160)2(492745)2(577500)2(121225)2(324660)2(499505)2340213860000 2, 17, 2269
05359260480247225238140205920 (80808)2(252967)2(245700)2(343265)2(221208)2(326183)2(320580)2160233906625 5, 29, 37, 7069
255110151415111510 (123)2(1515)2(1515)2(2139)2(1515)2(2139)2(2139)26855417 761713
6165330454245331490 (369)2(4545)2(4545)2(6417)2(1535)2(4783)2(4783)243397953 6329, 6857
822044060566044505 (492)2(6060)2(6060)2(8556)2(705)2(6081)2(6081)273460097 8162233
63294378686541323 (483)2(749)2(387)2(691)2(1407)2(1519)2(1377)22223841 2223841
110160019367480484038423 (2514)2(7650)2(5214)2(8910)2(38505)2(39177)2(38775)21555702929 17, 9337
1386902645685963666330 (2737)2(6895)2(6895)2(9359)2(6895)2(9359)2(9359)2127640737 2604913
156429085801180923535117780 (9594)2(118170)2(9281)2(118145)2(118170)2(166842)2(118145)227830345089 809, 34400921
216552612730813299416 (852)2(7332)2(1479)2(7431)2(9452)2(11932)2(9529)2143834161 17, 19, 445307
2167035781210800160659716 (10515)2(12891)2(17865)2(19359)2(14315)2(16141)2(20335)2469124881 83, 5652107
4325761092265227041521 (1308)2(2748)2(2948)2(3812)2(1959)2(3111)2(3289)216658161 241, 409
4487524100323801650768613 (12548)2(38756)2(11252)2(38356)2(15213)2(39699)2(14163)21545165801 3, 706523
4481232024640339136338464312995 (27552)2(339360)2(339360)2(479136)2(314205)2(461661)2(461661)2327536975817 23, 103, 271, 56687
704200204928041432012812824815 (53196)2(414804)2(137280)2(433680)2(58695)2(415545)2(139503)2188693631009 124058929
75612528584642429286516327956 (59796)2(243252)2(87549)2(251517)2(66004)2(244852)2(91901)264041767689 2843, 22526123
94556009072282601134028896 (10703)2(28825)2(14553)2(30465)2(30800)2(40804)2(32340)21762202016 2, 179, 34183
110413805520433602842826679 (5796)2(43396)2(28980)2(51860)2(27279)2(50929)2(39375)23400009825 136000393
1496411408228011324721130228285595 (92004)2(1133220)2(1133220)2(1599972)2(300045)2(1168653)2(1168653)22641472666793 31, 9467643967
1610442758855012187701216355823598 (99015)2(1219575)2(1219575)2(1721895)2(829527)2(1471623)2(1471623)23643233464529 359, 463, 2435393
17643752543924381529217208 (6839)2(24731)2(7791)2(25011)2(18433)2(30077)2(18807)2918553689 19, 596851
198030605610275552019612580 (6690)2(27795)2(21054)2(34221)2(14110)2(30445)2(24446)21325412841 229, 5787829
2898144905554514403913368650602 (57477)2(144795)2(144795)2(196539)2(76523)2(153355)2(153355)241179742521 31,863, 1539257
33029080518161024996142494661215290 (203073)2(2501265)2(2501265)2(3531489)2(295935)2(2510511)2(2510511)212517753438017 23, 60472238831
5936163240326480449355244846481093955 (365064)2(4496520)2(4496520)2(6348552)2(1153245)2(4627677)2(4627677)241500814802633 340297, 13550521
653617974035948049477524937948606989 (401964)2(4951020)2(4951020)2(6990252)2(727989)2(4988085)2(4988085)249232015950329 23, 237835825847
23744652960130592017974208179385924375820 (1460256)2(17986080)2(17986080)2(25394208)2(4612980)2(18510708)2(18510708)2664013036842128 340297, 13550521
641703208501229925318943529601903961474 (1272705)2(3206175)2(3206175)2(4351935)2(4160401)2(5095951)2(5095951)234628496708001 5783, 122203703
712083560401364820353924432848561653815 (1412292)2(3557820)2(3557820)2(4829244)2(2173615)2(3922769)2(3922769)226051631086497 17, 1532448887441

Contact information:

If you have any questions or contributions to the problem you can contact me at thomas.egense@gmail.com