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Order-6 magic cubes (the constant is 651)

   There exist order-6 magic cubes of classes simple, pantriagonal, and diagonal. An associated simple magic cube or an associated pantriagonal magic cube of order 6 can exist, but an associated diagonal magic cube of order 6 cannot exist.

an order-6 simple magic cube (non-associated) [W. Firth (-1889)]
Plane No.1
28134129186192
64130133190188
1821782124121125
1771812223126122
1441381741691610
1401421701731214
Plane No.2
53132135189187
17136131185191
1801841819127123
1831791720124128
1391411721751113
143137176171159
Plane No.3
1171141461526260
1181131501486458
5450109106168164
5256110105162166
154160706897102
156158667298101

Plane No.4
1201151491476357
1191161451516159
5155112107161165
5349111108167163
1551576571100103
153159696799104
Plane No.5
20620442457876
20220846417480
89931981993834
94901972003337
28308285212214
32268681216210
Plane No.6
20120748437379
20520344477775
95911931963640
92961941953935
31258883215209
27298487211213
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

This cube seems to be the first order-6 magic cube in the world.

an order-6 simple magic cube (associated) [Ingenieur Weidemann, 1922]
Plane No.1
18635213184321
30206190919125
199197222114198
193231516200204
71882820720912
3621832145211
Plane No.2
13911310610574114
1201011351189879
961281248712591
851229412913190
1028011713683133
1091077576140144
Plane No.3
175384014771180
434717117215266
541645158167157
168161575215855
611731541536248
150681786941145

Plane No.4
721761483914967
169155646344156
162591651605649
605015916653163
151654546170174
371467017717942
Plane No.5
7377141142110108
8413481100137115
127868812395132
126921309389121
138119998211697
10314311211110478
Plane No.6
6212334215181
20581018929210
131720120219424
192031961952018
192262082711187
21618533418231
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

An order-6 associated magic cube does exist though an order-6 associated magic square cannot exist.
This cube seems to be the first order-6 associated magic cube in the world.

an order-6 pantriagonal magic cube (non-associated, complete) [Gakuho Abe, 1948]
Plane No.1
114414198118176
140172190122189
180512610194136
2813559153109167
1311631451136336
1713211755149127
Plane No.2
141173192121177
178612512193137
214213197120177
1321641471126234
1693311657148128
2913358152111168
Plane No.3
161678847186102
661075118783157
103156182875271
20622793821393
21984221474202
94201209784326

Plane No.4
641085018982158
104154181865472
162689046185100
19994121673203
95199208774527
20723813721291
Plane No.5
105155183855370
160698948184101
651064918884159
96200210764425
20524803921192
20974021575204
Plane No.6
179412411195138
314315196119175
139174191123168
1703111556150129
3013460151110166
1301651461146135
the source: Akira Hirayama & Gakuho Abe, Researches in Magic Squares, Osaka Kyoikutosho, 1983, pp.165-166

This cube seems to be the first order-6 pantriagonal magic cube in the world.

an order-6 pantriagonal magic cube (associated) [Nakamura, February 2008]
Plane No.1
28214832158427
1561444714346115
410319410419551
1591415014049112
1691911620117210
1353016129160136
Plane No.2
63756218288181
106110719154192
60785918585184
13019313151866
165108164805579
12719612881839
Plane No.3
19997114420199
3915314612537151
94202411496204
1504212214914840
1241017920012612
4514715211943145

Plane No.4
72174986570172
20591173820793
17769689517567
1312120317615123
66180927164178
1181617320612018
Plane No.5
20834209892190
1381621375310952
21131212862487
3313232158139157
2516326110216111
3612935155142154
Plane No.6
81571885618782
710019710119848
105168771677658
1662211323114213
102171741707361
19013321343189
the source: original. Here is this cube of CSV format.

This cube seems to be the first order-6 associated pantriagonal magic cube in the world.

an order-6 diagonal magic cube (non-associated, concentric) [Walter Trump, September 2003]
Plane No.1
1091437612388112
871564917063126
140174521505382
756618251139138
136401486517686
1047214492132107
Plane No.2
137481576815883
15521982720762
341872121322183
14732120819370
442132318612173
134169601495980
Plane No.3
10310115936119133
16219620182955
46206281973171
163151911821054
931714211192124
841165818198114

Plane No.4
90565718417589
118211620918899
180111892021437
64202261997153
71200203625146
1281611603342127
Plane No.5
1021781179538121
502151919010167
12030520419597
111185216924106
17241943120545
9639100122179115
Plane No.6
1101258514573113
796116847154142
131431656716481
911513516678130
135177691524177
1059412974141108
the source: Walter Trump's site (http://www.trump.de/magic-squares/magic-cubes/cubes-1.html)

This cube is the first order-6 diagonal magic cube in the world.
The cube contains within it an order-4 horizontal-diagonal magic cube with the magic constant 434, which consists of integers from 1 to 32 and from 185 to 216.

an order-6 diagonal magic cube (non-associated, bordered) [Nakamura, July 2004]
Plane No.1
74815858171209
43216691472174
181142647415436
157315514165202
19731467221320
208169591594610
Plane No.2
352123816511190
2048210112013113
180951408911037
1891238012510628
1613411310087201
27517952206182
Plane No.3
187170681574524
164931389111253
4913211910281168
398899114133178
1912178127108198
193471496017230

Plane No.4
186175611505623
3212479126105185
148598115136203
1921291181038425
339613990109184
194421566716131
Plane No.5
29616055205196
167135116978650
4412277128107173
199941379211118
1918310411713026
212115716212188
Plane No.6
20740166661639
41114870215176
183751531436334
171446276152200
19521471145422
81775115154210
the source: original. Here is this cube of CSV format.

This cube is a diagonal magic cube, and it contains within it an order-4 horizontal-diagonal magic cube with the magic constant 434, which consists of consecutive integers from 77 to 140. So this cube is a bordered diagonal magic cube.
This cube is constructed by normalizing the order-6 sub cube of my order-16 bordered diagonal magic cube.
I have proved that a bordered diagonal (namely, consecutively concentric diagonal) magic cube can exist for any even order higher than 4. Here are examples for orders from 6 to 40.

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Order-7 magic cubes (the constant is 1204)

   There exist order-7 magic cubes of classes simple, pantriagonal, diagonal, and pandiagonal. Associated magic cubes of order 7 can exist for each of the classes. It is unknown whether an pantriagonal diagonal magic cube of order 7 can exist or not.

an order-7 simple magic cube (associated, horizontal-pandiagonal) [Abhinav Soni, 2001]
Plane No.1
7124316188304133249
1853081302467524020
2507224417189305127
2118630213124776241
1282517324518183306
2421518730313224877
3071292527423919184
Plane No.2
3321122775022244167
5421948164336109274
1683331062785122345
2755522049165330110
4616233410727952224
1112765622143166331
2184716333510828053
Plane No.3
2012319531114025678
3151372538219827192
7920224196312134257
1933091382548319928
2588020325190313135
2219431013925584200
1362598119726191314
Plane No.4
119284572292174339
226617134311628161
340113285582303175
622277172337117282
169341114286592314
283632281173338118
517034211528760225

Plane No.5
3015331814726385208
1442608920534150322
2093115431914126486
3161452619020635151
8721032148320142265
1523171462629120729
2668820433149321143
Plane No.6
291642369181297126
1317830112328868233
1202926523710182298
2341417929512428969
2991212936623811176
702358180296125290
1773001222946723212
Plane No.7
1603251052709221537
2679621241157329102
381613269927193216
1032689721342158323
2173915532710027294
3241042699821436159
9521140156328101273
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

an order-7 pantriagonal magic cube (associated) [John R. Hendricks, 1973]
Plane No.1
981341772202633066
139182218261304496
180223266302294137
221264307792135178
262305597140176219
303395138181224260
193136179222265308
Plane No.2
1041962322753181861
1942372803161659102
2352783212157100192
2763191962105190233
3171760103195238274
1558101193236279322
639919123427732020
Plane No.3
1592022943303073116
2002923353571114157
2903333376119155198
3313174117160203288
2972115158201293336
7711315619929133434
1181611972893323275
Plane No.4
2142573004985128171
2552984790133169212
2964588131174217253
4386129172215258301
9112717021325629948
1321752112542974689
1732162592954487130

Plane No.5
2693121255147183226
3101053145188231267
851143186229272315
5614118422727031313
1461892252683111154
187230273309952144
2282713141450142185
Plane No.6
3242467110153245281
2265108151243286329
7010614924128432727
1111542392823252568
1522442873232366109
2422853282864107150
2833262669112148240
Plane No.7
3679122165208251343
8412016320624934141
1251682042473393982
1662092523373780123
2072503424278121164
2483404083126162205
3383881124167210246
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

an order-7 diagonal magic cube (associated) [Walter Trump, 2003]
Plane No.1
2901261223369298176
2078532115035264142
5627716348220106334
1524312930818672251
3241609526710338217
173228711633723059
1393111978225419625
Plane No.2
1831924813330524076
1003282149227115742
341170632841136227
2581362220179315193
2245333116745274110
662941809237123295
3220414631815489261
Plane No.3
2129732615540269105
1781429212130023564
2719913431319184256
3361655127910843222
2461311724574303188
1443162098726615230
612821754225118339
Plane No.4
1123332195527616247
7125018521242128307
2631413420691320149
229583431721286115
1952425313831020381
3721610232315994273
2971826828912511232

Plane No.5
522611934016962283
3141927825713528200
156412709932721398
122301236652931798
8826015331210145317
2801094422352330166
2397530418918247132
Plane No.6
8325519026198140312
4922110733516450278
2347029917713291120
1512926514332220886
117338231602811743
3021877325213016244
2681043921196325161
Plane No.7
3191489026214733205
285114722857342171
1273062417724918420
9327215836215101329
1023812429618167288
2028030919423259137
1684627511133221854
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

Trump made this cube by using consecutive integers from 0 to 342. I converted the cube to the above normal magic cube.
This magic cube is diagonal but not pandiagonal. He says that he made it by transforming Frost's order-7 pandiagonal magic cube to destroy the sums in some pandiagonals.

an order-7 pandiagonal magic cube (associated) [A. H. Frost, 1866]
Plane No.1
327419899156213270
5210916622328033044
169226283340562119
293301865122179236
1875132189239247304
1351922002573142878
2102603173188145153
Plane No.2
113170227284341663
237294295966123180
3051976133183240248
7913619320125831522
1542042613183289146
2713284292100157214
4553110167224274331
Plane No.3
2493062077127184241
2380137194202259309
1471482052623193390
2152723293693101158
3324654111168218275
571141712282853427
1812382882961067124
Plane No.4
9114114920626332034
1592162733233794102
2763334755112162219
158115172229286343
1251822322892971168
2422503072171128185
3102481138195203253

Plane No.5
2202773344856106163
337259116173230287
6912617623329029812
1862432513081572129
2543112582139196197
3585142150207264321
1031602172673243895
Plane No.6
1370120177234291299
1301872442523021673
1982553122683140190
3222986143151208265
9610416121126832539
1642212783354950107
281338360117174231
Plane No.7
1911992563132784134
2663163087144152209
4097105155212269326
1081652222793364351
225282339461118175
3001464121178235292
7413118824524630317
the source: Harvey D. Heinz's site (http://www.magic-squares.net/)

This cube is an associated pandiagonal magic cube.
This cube seems to be the first order-7 magic cube in the world.

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This page was last updated on August 8, 2016.
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