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I think, I have a proof, that no pandiagonal semimagic (= "pan1,2agonal")
cube of order 8p+4 is possible. Please check it, since I'm not absolutely sure.
theorem 1: if A(-,-,-) is a (nonnormal) pan1,2agonal cube of order 4,
then A is constant, i.e. all entries A(-,-,-) are equal.
proof: by solving the system of linear equations with 144 equations
and 64 variables by computer.
corollary 2: if A(-,-,-) is a pan1,2agonal (not necessarily normal)
cube of order 4*p then all the 64 sums
S(x,y,z) = SUM [i,j,k=0..p-1] A(x+4*i,y+4*j,z+4*k)
for x,y,z=0..3 are equal.
proof: divide the cube orthogonally into p*p*p subcubes
B_(0..p-1,0..p-1,0..p-1) (-,-,-) of order 4 each.
Given x,y,z,D=(d1,d2,d3) ,
consider all the n*n pan1,2agonals of A into direction D
passing through (x+4*i,y+4*j,z+4*k) for some i,j,k in {0..p-1}.
They pass through each point of each subcube B in the corresponding
pan1,2agonal exactly once.
So the sums S(x,y,z) form a pan1,2agonal cube of order 4.
By theorem 1 they are all equal.
corollary 3: there is no normal pan1,2agonal cube of order m=8*p+4
proof: the sum of all elements in such a cube were (m^3-1)*m^3/3 .
By corollary 2 we can partition the entries into 64 sets of size
(2*p+1)^3 which all have the same sum.
But (m^3-1)*m^3/2 is not divisible by 64.
illustration of corollary 2 for example with m=12:
27 subcubes:
B_(0,0,0) = A---
B_(0,0,1) = B---
...
...
B_(2,2,1) = Z---
B_(2,2,2) = [---
64 sums:
S(0,0,0)=A000+B000+...+Z000+[000
...
...
S(3,3,3)=A333+B333+...+Z333+[333
12^3 entries:
A000 A001 A002 A003 B000 B001 B002 B003 C000 C001 C002 C003
A010 A011 A012 A013 B010 B011 B012 B013 C010 C011 C012 C013
A020 A021 A022 A023 B020 B021 B022 B023 C020 C021 C022 C023
A030 A031 A032 A033 B030 B031 B032 B033 C030 C031 C032 C033
D000 D001 D002 D003 E000 E001 E002 E003 F000 F001 F002 F003
D010 D011 D012 D013 E010 E011 E012 E013 F010 F011 F012 F013
D020 D021 D022 D023 E020 E021 E022 E023 F020 F021 F022 F023
D030 D031 D032 D033 E030 E031 E032 E033 F030 F031 F032 F033
G000 G001 G002 G003 H000 H001 H002 H003 I000 I001 I002 I003
G010 G011 G012 G013 H010 H011 H012 H013 I010 I011 I012 I013
G020 G021 G022 G023 H020 H021 H022 H023 I020 I021 I022 I023
G030 G031 G032 G033 H030 H031 H032 H033 I030 I031 I032 I033
A100 A101 A102 A103 B100 B101 B102 B103 C100 C101 C102 C103
A110 A111 A112 A113 B110 B111 B112 B113 C110 C111 C112 C113
A120 A121 A122 A123 B120 B121 B122 B123 C120 C121 C122 C123
A130 A131 A132 A133 B130 B131 B132 B133 C130 C131 C132 C133
D100 D101 D102 D103 E100 E101 E102 E103 F100 F101 F102 F103
D110 D111 D112 D113 E110 E111 E112 E113 F110 F111 F112 F113
D120 D121 D122 D123 E120 E121 E122 E123 F120 F121 F122 F123
D130 D131 D132 D133 E130 E131 E132 E133 F130 F131 F132 F133
G100 G101 G102 G103 H100 H101 H102 H103 I100 I101 I102 I103
G110 G111 G112 G113 H110 H111 H112 H113 I110 I111 I112 I113
G120 G121 G122 G123 H120 H121 H122 H123 I120 I121 I122 I123
G130 G131 G132 G133 H130 H131 H132 H133 I130 I131 I132 I133
A200 A201 A202 A203 B200 B201 B202 B203 C200 C201 C202 C203
A210 A211 A212 A213 B210 B211 B212 B213 C210 C211 C212 C213
A220 A221 A222 A223 B220 B221 B222 B223 C220 C221 C222 C223
A230 A231 A232 A233 B230 B231 B232 B233 C230 C231 C232 C233
D200 D201 D202 D203 E200 E201 E202 E203 F200 F201 F202 F203
D210 D211 D212 D213 E210 E211 E212 E213 F210 F211 F212 F213
D220 D221 D222 D223 E220 E221 E222 E223 F220 F221 F222 F223
D230 D231 D232 D233 E230 E231 E232 E233 F230 F231 F232 F233
G200 G201 G202 G203 H200 H201 H202 H203 I200 I201 I202 I203
G210 G211 G212 G213 H210 H211 H212 H213 I210 I211 I212 I213
G220 G221 G222 G223 H220 H221 H222 H223 I220 I221 I222 I223
G230 G231 G232 G233 H230 H231 H232 H233 I230 I231 I232 I233
A300 A301 A302 A303 B300 B301 B302 B303 C300 C301 C302 C303
A310 A311 A312 A313 B310 B311 B312 B313 C310 C311 C312 C313
A320 A321 A322 A323 B320 B321 B322 B323 C320 C321 C322 C323
A330 A331 A332 A333 B330 B331 B332 B333 C330 C331 C332 C333
D300 D301 D302 D303 E300 E301 E302 E303 F300 F301 F302 F303
D310 D311 D312 D313 E310 E311 E312 E313 F310 F311 F312 F313
D320 D321 D322 D323 E320 E321 E322 E323 F320 F321 F322 F323
D330 D331 D332 D333 E330 E331 E332 E333 F330 F331 F332 F333
G300 G301 G302 G303 H300 H301 H302 H303 I300 I301 I302 I303
G310 G311 G312 G313 H310 H311 H312 H313 I310 I311 I312 I313
G320 G321 G322 G323 H320 H321 H322 H323 I320 I321 I322 I323
G330 G331 G332 G333 H330 H331 H332 H333 I330 I331 I332 I333
J000 J001 J002 J003 K000 K001 K002 K003 L000 L001 L002 L003
J010 J011 J012 J013 K010 K011 K012 K013 L010 L011 L012 L013
J020 J021 J022 J023 K020 K021 K022 K023 L020 L021 L022 L023
J030 J031 J032 J033 K030 K031 K032 K033 L030 L031 L032 L033
M000 M001 M002 M003 N000 N001 N002 N003 O000 O001 O002 O003
M010 M011 M012 M013 N010 N011 N012 N013 O010 O011 O012 O013
M020 M021 M022 M023 N020 N021 N022 N023 O020 O021 O022 O023
M030 M031 M032 M033 N030 N031 N032 N033 O030 O031 O032 O033
P000 P001 P002 P003 Q000 Q001 Q002 Q003 R000 R001 R002 R003
P010 P011 P012 P013 Q010 Q011 Q012 Q013 R010 R011 R012 R013
P020 P021 P022 P023 Q020 Q021 Q022 Q023 R020 R021 R022 R023
P030 P031 P032 P033 Q030 Q031 Q032 Q033 R030 R031 R032 R033
J100 J101 J102 J103 K100 K101 K102 K103 L100 L101 L102 L103
J110 J111 J112 J113 K110 K111 K112 K113 L110 L111 L112 L113
J120 J121 J122 J123 K120 K121 K122 K123 L120 L121 L122 L123
J130 J131 J132 J133 K130 K131 K132 K133 L130 L131 L132 L133
M100 M101 M102 M103 N100 N101 N102 N103 O100 O101 O102 O103
M110 M111 M112 M113 N110 N111 N112 N113 O110 O111 O112 O113
M120 M121 M122 M123 N120 N121 N122 N123 O120 O121 O122 O123
M130 M131 M132 M133 N130 N131 N132 N133 O130 O131 O132 O133
P100 P101 P102 P103 Q100 Q101 Q102 Q103 R100 R101 R102 R103
P110 P111 P112 P113 Q110 Q111 Q112 Q113 R110 R111 R112 R113
P120 P121 P122 P123 Q120 Q121 Q122 Q123 R120 R121 R122 R123
P130 P131 P132 P133 Q130 Q131 Q132 Q133 R130 R131 R132 R133
J200 J201 J202 J203 K200 K201 K202 K203 L200 L201 L202 L203
J210 J211 J212 J213 K210 K211 K212 K213 L210 L211 L212 L213
J220 J221 J222 J223 K220 K221 K222 K223 L220 L221 L222 L223
J230 J231 J232 J233 K230 K231 K232 K233 L230 L231 L232 L233
M200 M201 M202 M203 N200 N201 N202 N203 O200 O201 O202 O203
M210 M211 M212 M213 N210 N211 N212 N213 O210 O211 O212 O213
M220 M221 M222 M223 N220 N221 N222 N223 O220 O221 O222 O223
M230 M231 M232 M233 N230 N231 N232 N233 O230 O231 O232 O233
P200 P201 P202 P203 Q200 Q201 Q202 Q203 R200 R201 R202 R203
P210 P211 P212 P213 Q210 Q211 Q212 Q213 R210 R211 R212 R213
P220 P221 P222 P223 Q220 Q221 Q222 Q223 R220 R221 R222 R223
P230 P231 P232 P233 Q230 Q231 Q232 Q233 R230 R231 R232 R233
J300 J301 J302 J303 K300 K301 K302 K303 L300 L301 L302 L303
J310 J311 J312 J313 K310 K311 K312 K313 L310 L311 L312 L313
J320 J321 J322 J323 K320 K321 K322 K323 L320 L321 L322 L323
J330 J331 J332 J333 K330 K331 K332 K333 L330 L331 L332 L333
M300 M301 M302 M303 N300 N301 N302 N303 O300 O301 O302 O303
M310 M311 M312 M313 N310 N311 N312 N313 O310 O311 O312 O313
M320 M321 M322 M323 N320 N321 N322 N323 O320 O321 O322 O323
M330 M331 M332 M333 N330 N331 N332 N333 O330 O331 O332 O333
P300 P301 P302 P303 Q300 Q301 Q302 Q303 R300 R301 R302 R303
P310 P311 P312 P313 Q310 Q311 Q312 Q313 R310 R311 R312 R313
P320 P321 P322 P323 Q320 Q321 Q322 Q323 R320 R321 R322 R323
P330 P331 P332 P333 Q330 Q331 Q332 Q333 R330 R331 R332 R333
S000 S001 S002 S003 T000 T001 T002 T003 U000 U001 U002 U003
S010 S011 S012 S013 T010 T011 T012 T013 U010 U011 U012 U013
S020 S021 S022 S023 T020 T021 T022 T023 U020 U021 U022 U023
S030 S031 S032 S033 T030 T031 T032 T033 U030 U031 U032 U033
V000 V001 V002 V003 W000 W001 W002 W003 X000 X001 X002 X003
V010 V011 V012 V013 W010 W011 W012 W013 X010 X011 X012 X013
V020 V021 V022 V023 W020 W021 W022 W023 X020 X021 X022 X023
V030 V031 V032 V033 W030 W031 W032 W033 X030 X031 X032 X033
Y000 Y001 Y002 Y003 Z000 Z001 Z002 Z003 [000 [001 [002 [003
Y010 Y011 Y012 Y013 Z010 Z011 Z012 Z013 [010 [011 [012 [013
Y020 Y021 Y022 Y023 Z020 Z021 Z022 Z023 [020 [021 [022 [023
Y030 Y031 Y032 Y033 Z030 Z031 Z032 Z033 [030 [031 [032 [033
S100 S101 S102 S103 T100 T101 T102 T103 U100 U101 U102 U103
S110 S111 S112 S113 T110 T111 T112 T113 U110 U111 U112 U113
S120 S121 S122 S123 T120 T121 T122 T123 U120 U121 U122 U123
S130 S131 S132 S133 T130 T131 T132 T133 U130 U131 U132 U133
V100 V101 V102 V103 W100 W101 W102 W103 X100 X101 X102 X103
V110 V111 V112 V113 W110 W111 W112 W113 X110 X111 X112 X113
V120 V121 V122 V123 W120 W121 W122 W123 X120 X121 X122 X123
V130 V131 V132 V133 W130 W131 W132 W133 X130 X131 X132 X133
Y100 Y101 Y102 Y103 Z100 Z101 Z102 Z103 [100 [101 [102 [103
Y110 Y111 Y112 Y113 Z110 Z111 Z112 Z113 [110 [111 [112 [113
Y120 Y121 Y122 Y123 Z120 Z121 Z122 Z123 [120 [121 [122 [123
Y130 Y131 Y132 Y133 Z130 Z131 Z132 Z133 [130 [131 [132 [133
S200 S201 S202 S203 T200 T201 T202 T203 U200 U201 U202 U203
S210 S211 S212 S213 T210 T211 T212 T213 U210 U211 U212 U213
S220 S221 S222 S223 T220 T221 T222 T223 U220 U221 U222 U223
S230 S231 S232 S233 T230 T231 T232 T233 U230 U231 U232 U233
V200 V201 V202 V203 W200 W201 W202 W203 X200 X201 X202 X203
V210 V211 V212 V213 W210 W211 W212 W213 X210 X211 X212 X213
V220 V221 V222 V223 W220 W221 W222 W223 X220 X221 X222 X223
V230 V231 V232 V233 W230 W231 W232 W233 X230 X231 X232 X233
Y200 Y201 Y202 Y203 Z200 Z201 Z202 Z203 [200 [201 [202 [203
Y210 Y211 Y212 Y213 Z210 Z211 Z212 Z213 [210 [211 [212 [213
Y220 Y221 Y222 Y223 Z220 Z221 Z222 Z223 [220 [221 [222 [223
Y230 Y231 Y232 Y233 Z230 Z231 Z232 Z233 [230 [231 [232 [233
S300 S301 S302 S303 T300 T301 T302 T303 U300 U301 U302 U303
S310 S311 S312 S313 T310 T311 T312 T313 U310 U311 U312 U313
S320 S321 S322 S323 T320 T321 T322 T323 U320 U321 U322 U323
S330 S331 S332 S333 T330 T331 T332 T333 U330 U331 U332 U333
V300 V301 V302 V303 W300 W301 W302 W303 X300 X301 X302 X303
V310 V311 V312 V313 W310 W311 W312 W313 X310 X311 X312 X313
V320 V321 V322 V323 W320 W321 W322 W323 X320 X321 X322 X323
V330 V331 V332 V333 W330 W331 W332 W333 X330 X331 X332 X333
Y300 Y301 Y302 Y303 Z300 Z301 Z302 Z303 [300 [301 [302 [303
Y310 Y311 Y312 Y313 Z310 Z311 Z312 Z313 [310 [311 [312 [313
Y320 Y321 Y322 Y323 Z320 Z321 Z322 Z323 [320 [321 [322 [323
Y330 Y331 Y332 Y333 Z330 Z331 Z332 Z333 [330 [331 [332 [333
9 pandiagonals in direction (0,1,1) starting
at sub-positions (1,2,2)
.. .. .. .. .. .. .. .. .. .. .. ..
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.. .. A102 .. .. .. B102 .. .. .. C102 ..
.. .. .. A113 .. .. .. B113 .. .. .. C113
A120 .. .. .. B120 .. .. .. C120 .. .. ..
.. A131 .. .. .. B131 .. .. .. C131 .. ..
.. .. D102 .. .. .. E102 .. .. .. F102 ..
.. .. .. D113 .. .. .. E113 .. .. .. F113
D120 .. .. .. E120 .. .. .. F120 .. .. ..
.. D131 .. .. .. E131 .. .. .. F131 .. ..
.. .. G102 .. .. .. H102 .. .. .. I102 ..
.. .. .. G113 .. .. .. H113 .. .. .. I113
G120 .. .. .. H120 .. .. .. I120 .. .. ..
.. G131 .. .. .. H131 .. .. .. I131 .. ..
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.. .. J102 .. .. .. K102 .. .. .. L102 ..
.. .. .. J113 .. .. .. K113 .. .. .. L113
J120 .. .. .. K120 .. .. .. L120 .. .. ..
.. J131 .. .. .. K131 .. .. .. L131 .. ..
.. .. M102 .. .. .. N102 .. .. .. O102 ..
.. .. .. M113 .. .. .. N113 .. .. .. O113
M120 .. .. .. N120 .. .. .. O120 .. .. ..
.. M131 .. .. .. N131 .. .. .. O131 .. ..
.. .. P102 .. .. .. Q102 .. .. .. R102 ..
.. .. .. P113 .. .. .. Q113 .. .. .. R113
P120 .. .. .. Q120 .. .. .. R120 .. .. ..
.. P131 .. .. .. Q131 .. .. .. R131 .. ..
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.. .. .. .. .. .. .. .. .. .. .. ..
.. .. S102 .. .. .. T102 .. .. .. U102 ..
.. .. .. S113 .. .. .. T113 .. .. .. U113
S120 .. .. .. T120 .. .. .. U120 .. .. ..
.. S131 .. .. .. T131 .. .. .. U131 .. ..
.. .. V102 .. .. .. W102 .. .. .. X102 ..
.. .. .. V113 .. .. .. W113 .. .. .. X113
V120 .. .. .. W120 .. .. .. X120 .. .. ..
.. V131 .. .. .. W131 .. .. .. X131 .. ..
.. .. Y102 .. .. .. Z102 .. .. .. [102 ..
.. .. .. Y113 .. .. .. Z113 .. .. .. [113
Y120 .. .. .. Z120 .. .. .. [120 .. .. ..
.. Y131 .. .. .. Z131 .. .. .. [131 .. ..
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sum of the 9 pandiagonals = S(1,0,2) + S(1,1,3) + S(1,2,0) + S(1,3,1)
the sums S(-,-,-) form a pandiagonal (not normal) 4-cube,
when the above 12-cube is pandiagonal.
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