Magic Series
Magic series for magic cubes and hypercubes
This page is dedicated to my Canadian friend John Hendricks. Most of our knowledge about magic hypercubes was discovered by him.
He suggested to use n for the dimension and m for the order of a hypercube.
There are cubes of order 5 where all straight lines are magic. You may call them strictly magic cubes. Such cubes have got 75 monagonals (= 25 rows + 25 columns + 25 pilars), 30 diagonals (= short diagonals) and 4 triagonals (= long diagonals).
Each of these 109 lines is magic. The 5 numbers in each line sum up to 315.
If the result of a sum of 5 distinct positive integers, each not greater than 125, equals 315
then this sum is called a magic cube series of order 5.
For magic cube series of order m we demand:
- Number of terms in the sum: m
- Maximal value of an integer: m3
- Value of the sum: (m3 + 1)·m/2

This concept may be used for hypercubes of any dimension n and order m:
- Number of terms in the sum: m
- Maximal value of an integer: mn
- Value of the sum: (mn + 1)·m/2

Using the same algorithm as for magic squares series, I got the following numbers of magic series (2005-02-12).

Number of magic series for hypercubes of dimension n and order m


Dimension		 Order
m = 1		 Order
m = 2		 Order
m = 3		 Order
m = 4		 Order
m = 5		 Order
m = 6		 Order
m = 7
n = 1 1 1 1 1 1 1 1
n = 2 1 2 8 86 1394 32134 957332
n = 3 1 4 85 6786 1142341 338832214 156623626331
n = 4 1 8 800 457906 751910094 2766058729144  19279934097075600
n = 5 1 16 7321 29695346  474791829591  21682691008200124 (*)
n = 6 1 32 66248  1906778226 (*) (*) (*)
n = 7 1 64  597325  122134408306 (*) (*) (*)
Formula 1 2n-1 Nn(3) Nn(4) (*) (*) (*)

Order-3 formula:  Nn(3) = (3n - 1)2 / 8 + a   with a = 0 for even n   and a = 1/2 for odd n

Order-4 formula:  Nn(4) = (2u3 - 9u2 + 18u + 16) / 72   withu = 4n (= max. number in hypercube of dimension n)

Both formula were found on February 16th, 2005.
Nn(4) was found experimentally, whereas Nn(3) was proved (not elegantly but rather complicated).
(*) In summer 2006 I determined formulae for the orders 5 to 11: Next page
Number of magic series for hypercubes of dimension 3
(= Number of magic cube series) 
   N3(01) = 1
   N3(02) = 4
   N3(03) = 85
   N3(04) = 6786
   N3(05) = 1142341
   N3(06) = 338832214
   N3(07) = 156623626331
   N3(08) = 104510988949316
   N3(09) = 95268144607230087
   N3(10) = 113890197280403493542
   N3(11) = 173010424861377562731014
   N3(12) = 325702485631908523866475222
   N3(13) = 744530535411231865659859497116
   N3(14) = 2032002010218114238831528413724970
   N3(15) = 6527910014315748187894308484050347987
   N3(16) = 24387539775140714873044335703400231155136
   N3(17) = 104848714411898862472408574437293947290702029
   N3(18) = 514044750180549050442567090101496413473966382586
   N3(19) = 2851028862119546169537090377842408485779881321936932
   N3(20) = 17761622824968263447480846131961889702954933930143640222
   N3(21) = 123507908155489415259714919787990912099853019354696330382798
   N3(22) = 953171345162459056079884898190780075022378458290023885264280280
   N3(23) = 8122418609655059149917607124169919389170847292056282240224982459626
   N3(24) = 76071078142263489144271535348244319093125085424867933793021114142686578
   N3(25) = 779713664953507723219864299512253552154552656079012342466306984743947563650
   N3(26) = 8712657247151011533667805994365535615436854780746585820112459443349827000311924
   N3(27) = 105760216311436165183093905738108357978036899914057350653021726339429929822378954480
   N3(28) = 1390042617972072415527107819834364205977961640723184756497410212317708290067795924300716
   N3(29) = 19722122087194379309353177795515784954079133716100546130420206136600289567836012674671474866
   N3(30) = 301217770585949430821677726637765162566699227115283269125110601447255945201684345965851046031534
   N3(31) = 4939426566399261462756076081272647860686256065099788060372415751105529825052176969165454985576861687
   N3(32) = 8675333889314245080115100105795164944345175600520457112625490684274074817000120194445203249053716751
            8874
   N3(33) = 1628260383813869207998922142039476022036287910014547576454513639913242413797043349921234206115039249
            029401559
   N3(34) = 3258867359557494742040012044191360271780776833323501077374328036372135471813127804372905190369023718
            3571482153844
   N3(35) = 6941429955900658689354188844561486663268998734833537572536847347647772016708628487119214577730510848
            14223437931760829
   N3(36) = 1570568428641886366948015262874480318427961799771800848140536925691269867264395917355337998996209592
            1083958005905871514746
   N3(37) = 3768114474329823826299675817021952584305940597993146395470748371760538024101231176029559084496053034
            90089692439321097280968348
   N3(38) = 9570346906458367638793398255351711173255772809159197130577281447291472048909951274939589720919764528
            599171987491881597483497978056
   N3(39) = 2569120821555720707700542468344195880287696558391464567715117453885076390575874772987559433933441916
            33967279223746800013114750890957995
   N3(40) = 7278605055911668334821326725993341386246010067035229244115656361166847064174969690779723955339042061
            538446483745723106534260336145395931754
   N3(41) = 2173230779213848713536695719675543557861326443189198808084324465855241186783613395813072181858966735
            18037723537283407054157185125937526884924547
   N3(42) = 6829298411921338006682867206461792760412418988361589425869202105922011693612718080319580419149794264
            506973600454773215852929617378344144218139843824
   N3(43) = 2255829345518394586285834583584524474629406046712981866372906733588621009189447252437345689625845578
            89452984034625257951893900947585591524987253212582406
   N3(44) = 7822947984413726559094779131725299808529220978220624834706456876701893988167444602361210917017816391
            244014741950061011673276914420498487689389377679974192876
   N3(45) = 2844904642704956862508680330434354929244711824913316419924930601010110970725088345988089565096916364
            94755258460051892130689339658828802896364370951497532824714454
   N3(46) = 1083727861587641336710995329753513129605879455959676060092481969704257882604072398074152992702161914
            9097277308360159892159164917901117506631287598885629149441267071878
   N3(47) = 4319881088507975829734230068822030067860139202912025731928318359364488659389416303018655828090856628
            49226845895553833223861380722128636160327965907150014469783239765091690
   N3(48) = 1800057232871625132978552441397348100338861819821751135488500215455500164316531542021202857166758771
            0051235878239481057938863848598002175495581689104574656199947129450455741708
   N3(49) = 7833333191430341278522787494348181579391407349629970428943064607841593488692410810837146963661921532
            47802155657625782308681281586213051813161391603678207726006497112196985676654724
   N3(50) = 3556748105603989234823119461374755695798529979798571501566339610225751331578675310920357327105610726
            2293749404075323047302623168999163662007179828837967313311350537310758255886452496692

   N3(100)= 1471350942827991124132253858550690876482115191673038840314670850799568188033396757689961539703310655
            4380072163888531628289538043515224898124360560826499542710495764658104038928328140558424407620051866
            5039086551579035293327528828557245522946934356819877526113136664678366937360806224281040278522557304
            3081638034089207192136464852931347472144009117175208822482012115577686696977857267865190895031912398
            192433455025101020359249399781677698

   N3(150)= 1015058984725358739400257699818044218023590596485541978713833627223210811651338257677296530889713903
            0056093263627800617386845914271944516628615329076983794334069831210768258772763395118597863672331257
            8081021439380085010695658833415355043182608965341965982231493494519258236664210511758828477548219439
            0359878244829930520391840397393259870046877174070410904514819520247123687477971530444393867641202786
            6233914139092615808379867838855469633644757645585704994563927304798131046810102121199940111511015758
            0540246955004876030039159587357436238442923397797383805993529121997826170159936169994278743238138031
            8284031879995332764948393543955193488655364806381579931914169711745131792809099172521836335784741529
            7714356686

   N3(200)= 6406245515885694440138075060451214459650333203982281354698031613068195252070201317013328004209162979
            8147607141028776082434192614145168526114479513723186013887408111014287453788219428114789661865946880
            3458876399192479599647187147576537407906306856719146722519601563362195782957743365530014342436292986
            7467830231111424425893316256767952066759575185445549422917928633818512778473246945148088978933440273
            6708056507696856453193293973221525059511328266987297938341065433394089023673929360114548642469594638
            7787064303242954037675534628462083078616193751998912705492276758132620502274957914655065984726726692
            5808560261999649572546512749094943457513047606293554574800040591300324848141533352749661591563184577
            2519068419390512594327344076351949372879228808236366803784197802581313165018575960665119890856996186
            6186078912160892998908246882678458914582195423246398826274992491065690940477925067198639656527401462
            89715573235066384525755138302440920995385958790567136979639900848174778076112927531019709760630608
Terms 17 to 30 were added from 2007-01-04 to 2007-02-03 (calculated with Gerbicz's algorithm)
Later 50 terms were calculated by a common PC (3 GHz, 1 GB RAM) within two days.
All numbers for orders > 50 were calculated in 2013 with the method of Dirk Kinnaes.
Number of magic series for hypercubes of dimension 4 
   N4(01) = 1
   N4(02) = 8
   N4(03) = 800
   N4(04) = 457906
   N4(05) = 751910094
   N4(06) = 2766058729144
   N4(07) = 19279934097075600
   N4(08) = 228594811277217786320
   N4(09) = 4264819655248960858342656
   N4(10) = 118132052487666384802213007240
   N4(11) = 4643768453772212643643327799069020
   N4(12) = 249910611212426874400489215388031291754
   N4(13) = 17879943059627473884359465048565343208306660
   N4(14) = 1659694058087666324322349501564041153231721074328
   N4(15) = 195811840766424991031393098828021058542782512606563208
   N4(16) = 28851267146351983233823608598248115643943930325659092454940
   N4(17) = 5228863104684345825174562208556576509465064597808542674228301378
   N4(18) = 1150288132762271809213881969579045654302729771403333915087698909956912
   N4(19) = 303593807909897913226526653065876795977788518791081663327003953776116575660
   N4(20) = 95141304887696226735982367947818131861125408699521534114572649089416273825549048

   N4(30) = 2098638659424343929327316561091573489925102531006550724812628752340187131974624333952158505430765286
            485571149136065994945813284537113976552

   N4(40) = 2225463558912637234807260241554796228893871604172108460624694732515941880265526830600236805068402592
            8025093660720301317419254244693748705538190621768225321200399930786354775578799833314723279181723540
            40

   N4(50) = 6377807557182457458433629410686306418947568052088945549980823082854078022580506297909206313147886341
            6952960493065091011155185114641571623893990954342073670719051174001062857472820939973951748152653735
            77344974657558034059241082347764831515486555758377889243675145798552

   N4(60) = 9414526840971139887814151681637325317695253710633867479367536638554935700222892778817086281615458462
            6741769457389126663904027273882542090488411057692537059371585744661978880571374628136652670775780750
            7861785479430462704589609523364725232206355974587170826818774684367642411906077843912127546708443036
            2309821361962877715567855198967463430

   N4(70) = 2430132585092435367573576883250941569683824759504088489054715861626277755775378566116327097323815015
            3083355815728961104790804033242906665754237437194987911406080194400167883121175788284160082106663486
            6929696068318173122028716599376608220277259038441702314925034449533244248051504442135633141121822732
            7415812744360182984881784177385725130670505655908557061732895117770523452883068260563423201274884052
            977470322
Terms 10 to 20 were added from 2007-01-06 to 2007-01-28 (calculated with Gerbicz's algorithm)
All numbers for m > 20 were calculated in 2013 with Dirk Kinnaes' program.
Read more about magic hypercubes on the sites of
John Hendricks and Aale de Winkel. (external links)
Back to the Table of Contents
Walter Trump, Nürnberg, Germany, (c) 2005-02-08 (last modified: 2013-06-13)