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.
Thanks to Aale de Winkel who noticed some mistakes in previous versions of this page.
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) = 86753338893142450801151001057951649443451756005204571126254906842740748170001201944452032490537167518874
   N3(33) = 1628260383813869207998922142039476022036287910014547576454513639913242413797043349921234206115039249029401559
   N3(34) = 32588673595574947420400120441913602717807768333235010773743280363721354718131278043729051903690237183571482153844
   N3(35) = 694142995590065868935418884456148666326899873483353757253684734764777201670862848711921457773051084814223437931760829
   N3(36) = 15705684286418863669480152628744803184279617997718008481405369256912698672643959173553379989962095921083958005905871514746
   N3(37) = 376811447432982382629967581702195258430594059799314639547074837176053802410123117602955908449605303490089692439321097280968348
   N3(38) = 9570346906458367638793398255351711173255772809159197130577281447291472048909951274939589720919764528599171987491881597483497978056
   N3(39) = 256912082155572070770054246834419588028769655839146456771511745388507639057587477298755943393344191633967279223746800013114750890957995
   N3(40) = 7278605055911668334821326725993341386246010067035229244115656361166847064174969690779723955339042061538446483745723106534260336145395931754
   N3(41) = 217323077921384871353669571967554355786132644318919880808432446585524118678361339581307218185896673518037723537283407054157185125937526884924547
   N3(42) = 6829298411921338006682867206461792760412418988361589425869202105922011693612718080319580419149794264506973600454773215852929617378344144218139843824
   N3(43) = 225582934551839458628583458358452447462940604671298186637290673358862100918944725243734568962584557889452984034625257951893900947585591524987253212582406
   N3(44) = 7822947984413726559094779131725299808529220978220624834706456876701893988167444602361210917017816391244014741950061011673276914420498487689389377679974192876
   N3(45) = 284490464270495686250868033043435492924471182491331641992493060101011097072508834598808956509691636494755258460051892130689339658828802896364370951497532824714454
   N3(46) = 10837278615876413367109953297535131296058794559596760600924819697042578826040723980741529927021619149097277308360159892159164917901117506631287598885629149441267071878
   N3(47) = 431988108850797582973423006882203006786013920291202573192831835936448865938941630301865582809085662849226845895553833223861380722128636160327965907150014469783239765091690
   N3(48) = 18000572328716251329785524413973481003388618198217511354885002154555001643165315420212028571667587710051235878239481057938863848598002175495581689104574656199947129450455741708
   N3(49) = 783333319143034127852278749434818157939140734962997042894306460784159348869241081083714696366192153247802155657625782308681281586213051813161391603678207726006497112196985676654724
   N3(50) = 35567481056039892348231194613747556957985299797985715015663396102257513315786753109203573271056107262293749404075323047302623168999163662007179828837967313311350537310758255886452496692
Terms 17 to 30 were added from 2007-01-04 to 2007-02-03 (calculated with Gerbicz's algorithm)
All 50 terms were calculated by a common PC (3 GHz, 1 GB RAM) within two days.
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
Terms 10 to 20 were added from 2007-01-06 to 2007-01-28 (calculated with Gerbicz's algorithm)
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: 2007-02-03)