3486:
74:
33:
241:
176:
2339:
In case the hypercube also sum when all the numbers are raised to the power p one gets p-multimagic hypercubes. The above qualifiers are simply prepended onto the p-multimagic qualifier. This defines qualifications as {r-agonal 2-magic}. Here also "2-" is usually replaced by "bi", "3-" by "tri" etc.
2170:
Defined as the change of into alongside the given "axial"-direction. Equal permutation along various axes can be combined by adding the factors 2. Thus defining all kinds of r-agonal permutations for any r. Easy to see that all possibilities are given by the corresponding permutation of m numbers.
2335:
Note: This series doesn't start with 0 since a nill-agonal doesn't exist, the numbers correspond with the usual name-calling: 1-agonal = monagonal, 2-agonal = diagonal, 3-agonal = triagonal etc.. Aside from this the number correspond to the amount of "-1" and "1" in the corresponding pathfinder.
2454:
Caution: some people seems to equate {compact} with {compact} instead of {compact}. Since this introductory article is not the place to discuss these kind of issues I put in the dimensional pre-superscript to both these qualifiers (which are defined as shown) consequences of {compact} is that
2924:
Defined as the change of into alongside the given "axial"-direction. Equal permutation along various axes with equal orders can be combined by adding the factors 2. Thus defining all kinds of r-agonal permutations for any r. Easy to see that all possibilities are given by the corresponding
835:
Analogy suggest that in the higher dimensions we ought to employ the term nasik as implying the existence of magic summations parallel to any diagonal, and not restrict it to diagonals in sections parallel to the plane faces. The term is used in this wider sense throughout the present
2091:
Where reflect(k) true iff coordinate k is being reflected, only then 2 is added to R. As is easy to see, only n coordinates can be reflected explaining 2, the n! permutation of n coordinates explains the other factor to the total amount of "Aspectial variants"!
1463:
Further: without restrictions specified 'k' as well as 'i' run through all possible values, in combinations same letters assume same values. Thus makes it possible to specify a particular line within the hypercube (see r-agonal in pathfinder section)
1882:
1121:
1590:
1976:
Both methods fill the hypercube with numbers, the knight-jump guarantees (given appropriate vectors) that every number is present. The Latin prescription only if the components are orthogonal (no two digits occupying the same position)
2603:
Description of more general methods might be put here, I don't often create hyperbeams, so I don't know whether
Knightjump or Latin Prescription work here. Other more adhoc methods suffice on occasion I need a hyperbeam.
2062:
in the above equation, besides that on the result one can apply a manipulation to improve quality. Thus one can specify pe the J. R. Hendricks / M. Trenklar doubling. These things go beyond the scope of this article.
1906:(modular equations). This method is also specified by an n by n+1 matrix. However this time it multiplies the n+1 vector , After this multiplication the result is taken modulus m to achieve the n (Latin) hypercubes:
453:
1658:
further until (after m steps) a position is reached that is already occupied, a further vector is needed to find the next free position. Thus the method is specified by the n by n+1 matrix:
1625:
2261:
which specifies all (broken) r-agonals, p and q ranges could be omitted from this description. The main (unbroken) r-agonals are thus given by the slight modification of the above:
3121:
The hyperbeam that is usually added to change the here used "analytic" number range into the "regular" number range. Other constant hyperbeams are of course multiples of this one.
2055:
Most compounding methods can be viewed as variations of the above, As most qualifiers are invariant under multiplication one can for example place any aspectual variant of H
1735:
1416:
1252:
1009:
2365:}. The strange generalization of square 'perfect' to using it synonymous to {diagonal} in cubes is however also resolve by putting curly brackets around qualifiers, so {
1497:
860:
In 1939, B. Rosser and R. J. Walker published a series of papers on diabolic (perfect) magic squares and cubes. They specifically mentioned that these cubes contained 13
1127:
Note: The notation for position can also be used for the value on that position. Then, where it is appropriate, dimension and order can be added to it, thus forming:
2580:
Note: The notation for position can also be used for the value on that position. There where it is appropriate dimension and orders can be added to it thus forming:
1656:
1455:
1002:
850:
It is not difficult to perceive that if we push the Nasik analogy to higher dimensions the number of magic directions through any cell of a k-fold must be ½(3-1).
1724:
1267:
When a specific coordinate value is mentioned the other values can be taken as 0, which is especially the case when the amount of 'k's are limited using pe. #
2150:
Defined as the exchange of components, thus varying the factor m in m, because there are n component hypercubes the permutation is over these n components
3085:
of every hyperbeam with this normal, changing the source, changes the hyperbeam. Basic multiplications of normal hyperbeams play a special role with the
469:
3427:
2455:
several figures also sum since they can be formed by adding/subtracting order 2 sub-hyper cubes. Issues like these go beyond this articles scope.
2302:
Besides more specific qualifications the following are the most important, "summing" of course stands for "summing correctly to the magic sum"
706:
would apply to all dimensions of magic hypercubes in which the number of correctly summing paths (lines) through any cell of the hypercube is
2162:(usually denoted by ) is used with two dimensional matrices, in general though perhaps "coordinate permutation" might be preferable.
3298:
2916:(usually denoted by ) is used with two dimensional matrices, in general though perhaps "coördinaatpermutation" might be preferable.
1467:
Note: as far as I know this notation is not in general use yet(?), Hypercubes are not generally analyzed in this particular manner.
298:
280:
262:
222:
157:
60:
2879:, directions with equal orders contribute factors depending on the hyperbeam's orders. This goes beyond the scope of this article.
204:
91:
46:
251:
3307:
Planck, C., M.A., M.R.C.S., The Theory of Paths Nasik, 1905, printed for private circulation. Introductory letter to the paper
138:
3286:
95:
110:
3525:
2340:("1-magic" would be "monomagic" but "mono" is usually omitted). The sum for p-Multimagic hypercubes can be found by using
2240:
This gives 3 directions. since every direction is traversed both ways one can limit to the upper half of the full range.
2206:) since a component is filled with radix m digits, a permutation over m numbers is an appropriate manner to denote these.
831:. In 1905 Dr. Planck expanded on the nasik idea in his Theory of Paths Nasik. In the introductory to his paper, he wrote;
3475:
3420:
2912:
The exchange of coördinaat into , because of n coördinates a permutation over these n directions is required. The term
2158:
The exchange of coordinate into , because of n coordinates a permutation over these n directions is required. The term
117:
2612:
Amongst the various ways of compounding, the multiplication can be considered as the most basic of these methods. The
1985:
Amongst the various ways of compounding, the multiplication can be considered as the most basic of these methods. The
385:
2105:(explicitly stated here: the minimum of all corner points. The axial neighbour sequentially based on axial number)
3450:
1158:
between brackets, these cannot have the same value, though in undetermined order, which explains the equality of:
3530:
2448:
197:
191:
124:
2846:
Aspectial variants, which are obtained by coördinate reflection ( → ) effectively giving the
Aspectial variant:
2075:
Aspectial variants, which are obtained by coordinate reflection ( --> ) and coordinate permutations ( -->
3584:
3579:
3520:
3281:
Thomas R. Hagedorn, On the existence of magic n-dimensional rectangles, Discrete
Mathematics 207 (1999), 53-63.
1147:
are outside the range it is simply moved back into the range by adding or subtracting appropriate multiples of
730:
545:
519:
1595:
488:. Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by
3413:
563:
This definition of "perfect" assumes that one of the older definitions for perfect magic cubes is used. The
106:
84:
639:
is a magic hypercube with the added restriction that all possible lines through each cell sum correctly to
3656:
3510:
2511:
of a hyperbeam with the letter 'm' (appended with the subscripted number of the direction it applies to).
2341:
2471:) is a variation on a magic hypercube where the orders along each direction may be different. As such a
2354:
3086:
3078:
2933:
In case no restrictions are considered on the n-agonals a magic hyperbeam can be represented shown in
1877:{\displaystyle P_{k}=P_{0}+\sum _{l=0}^{n-1}((k\backslash m^{l})\ \%\ m)V_{l};\quad k=0\dots m^{n}-1.}
3372:
2444:
2095:
Aspectial variants are generally seen as being equal. Thus any hypercube can be represented shown in
52:
1116:{\displaystyle \left\langle {}_{k}i;\ k\in \{0,\cdots ,n-1\};\ i\in \{0,\cdots ,m-1\}\right\rangle }
3620:
3231:
Andrews, W. S., Magic
Squares and Cubes, Dover Publ. 1917. Essay pages 363-375 written by C. Planck
2871:
In case one views different orientations of the beam as equal one could view the number of aspects
2142:
Note: '#', '^', '_' and '=' are essential part of the notation and used as manipulation selectors.
1585:{\displaystyle \langle 1,2\rangle ,\langle 1,-2\rangle ,\langle -1,2\rangle ,\langle -1,-2\rangle }
3293:
Harvey D. Heinz & John R. Hendricks, Magic Square
Lexicon: Illustrated, self-published, 2000,
255:
that states a
Knowledge editor's personal feelings or presents an original argument about a topic.
3551:
3266:
3254:
3140:
3130:
2868:
Where reflect(k) true if and only if coordinate k is being reflected, only then 2 is added to R.
2495:
article in close detail, and just as that article serves merely as an introduction to the topic.
2348:
1276:
1163:
539:
1891:
3589:
3500:
3397:
3294:
3147:
1969:
often uses modular equation, conditions to make hypercubes of various quality can be found on
186:
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131:
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3090:
2955:
2876:
2492:
1634:
1421:
890:
3615:
3563:
3304:
J.R.Hendricks: Magic
Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9
489:
3367:
3320:
2904:
Note: '^' and '_' are essential part of the notation and used as manipulation selectors.
1486:
Besides more specific constructions two more general construction method are noticeable:
354:
such that the sums of the numbers on each pillar (along any axis) as well as on the main
3625:
3460:
873:
359:
355:
1664:
3650:
3455:
3436:
1898:
conditions to create with this method "Path Nasik" (or modern {perfect}) hypercubes.
601:. Nasik was defined in this manner by C. Planck in 1905. A nasik magic hypercube has
3635:
3605:
3515:
3465:
3352:
2484:
812:
327:
3485:
2243:
With these pathfinders any line to be summed over (or r-agonal) can be specified:
2451:, {compact complete} is the qualifier for the feature in more than 2 dimensions.
3630:
3610:
3377:
3082:
2989:
When the orders are not relatively prime the n-agonal sum can be restricted to:
2508:
1475:
824:
548:
diagonal also sum up to the hypercube's magic number, the hypercube is called a
312:
73:
3353:
History, definitions, and examples of perfect magic cubes and other dimensions.
3332:
2088:Σ ((reflect(k)) ? 2 : 0) ; perm(0..n-1) a permutation of 0..n-1
17:
3546:
3470:
3387:
3310:
Marián
Trenkler, Magic rectangles, The Mathematical Gazette 83(1999), 102-105.
3135:
2887:
Besides more specific manipulations, the following are of more general nature
2488:
2113:
Besides more specific manipulations, the following are of more general nature
1970:
331:
3244:, 1939. A bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4
2830:
since any number has but one complement only one of the directions can have m
2102:= min() (by reflection) < ; k = 0..n-2 (by coordinate permutation)
1494:
This construction generalizes the movement of the chessboard horses (vectors
3392:
3382:
3326:
3077:
This hyperbeam can be seen as the source of all numbers. A procedure called
2504:
320:
3378:
An ambitious ongoing work on classifications of magic cubes and tesseracts
3222:, 1905, printed for private circulation. Introductory letter to the paper.
2425:Σ is symbolic for summing all possible k's, there are 2 possibilities for
785:. He then demonstrated the concept with an order-7 cube we now class as
351:
3358:
An alternative definition of
Perfect, with history of recent discoveries
2218:", these directions are simplest denoted in a ternary number system as:
1460:(#j=n-1 can be left unspecified) j now runs through all the values in .
816:
3357:
2553:
Further: In this article the analytical number range is being used.
2199:
Usually being applied at component level and can be seen as given by
3383:
A variety of John R. Hendricks material, written under his direction
2184:
Further when all the axes undergo the same permutation (R = 2-1) an
855:
W. S. Andrews, Magic
Squares and Cubes, Dover Publ., 1917, page 366
2561:
in order to keep things in hand a special notation was developed:
2391:} : {all pairs halve an n-agonal apart sum equal (to (m - 1)}
1944:" (?i.e. Basic manipulation) are generally applied before these LP
884:
in order to keep things in hand a special notation was developed:
868:
pandiagonal magic squares parallel to the faces of the cube, and 6
828:
820:
515:= 1. A construction of a magic hypercube follows from the proof.
3405:
2188:
is achieved, In this special case the 'R' is usually omitted so:
1151:, as the magic hypercube resides in n-dimensional modular space.
745:
magic cube would have 13 magic lines passing through each of its
3346:
2347:
Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, the
761:
magic tesseract would have 40 lines passing through each of its
573:
possible lines sum correctly for the hypercube to be considered
3409:
2292:
with numbers in the analytical numberrange has the magic sum:
569:(John R. Hendricks) requires that for any dimension hypercube,
234:
169:
67:
26:
3368:
A Magic Hypercube encyclopedia with a broad range of material
3340:
2954:
Qualifying the hyperbeam is less developed then it is on the
741:
cells. This was A.H. Frost’s original definition of nasik. A
693:. This definition is the same as the Hendricks definition of
1801:
3009:
with all orders relatively prime this reaches its maximum:
2214:
J. R. Hendricks called the directions within a hypercubes "
473:
375:). If a magic hypercube consists of the numbers 1, 2, ...,
252:
personal reflection, personal essay, or argumentative essay
2958:
in fact only the k'th monagonal direction need to sum to:
841:
C. Planck, M.A., M.R.C.S., The Theory of Paths Nasik, 1905
526:, that will create magic hypercubes of any dimension with
2822:
Which goes beyond the scope of this introductory article
2742:
A fact that can be easily seen since the magic sums are:
2443:} is the "modern/alternative qualification" of what Dame
2317:} : all (unbroken and broken) r-agonals are summing.
1631:
and further numbers are sequentially placed at positions
2789:=1 of course, which allows for general identities like:
2369:} means {pan r-agonal; r = 1..n} (as mentioned above).
258:
2982:
for all k = 0..n-1 for the hyperbeam to be qualified {
2522: : the amount of directions within a hyperbeam.
1738:
1667:
1637:
1598:
1500:
1424:
1279:
1166:
1012:
893:
737:
square because 4 magic line pass through each of the
388:
2770:
is even, the product is even and thus the only way S
3598:
3572:
3539:
3493:
3443:
2436:for {complete} the complement of is at position .
2310:} : all main (unbroken) r-agonals are summing.
773:In 1866 and 1878, Rev. A. H. Frost coined the term
98:. Unsourced material may be challenged and removed.
2894: : coördinate permutation (n == 2: transpose)
2351:is technically seen {1-agonal 2-agonal 3-agonal}.
2126: : coordinate permutation (n == 2: transpose)
1876:
1718:
1650:
1619:
1584:
1449:
1410:
1246:
1139:runs through the dimensions, while the coordinate
1115:
996:
447:
3042:The following hyperbeams serve special purposes:
2491:and magic hypercube. This article will mimic the
846:In 1917, Dr. Planck wrote again on this subject.
699:, but different from the Boyer/Trump definition.
3284:Thomas R. Hagedorn, Magic rectangles revisited,
495:Marian Trenkler proved the following theorem: A
448:{\displaystyle M_{k}(n)={\frac {n(n^{k}+1)}{2}}}
358:are all the same. The common sum is called the
2380:} : {all order 2 subhyper cubes sum to 2 S
848:
833:
811:as a respect to the great Indian Mathematician
3373:A Unified classification system for hypercubes
3242:Magic Squares: Published papers and Supplement
2077:i]) effectively giving the Aspectial variant:
1143:runs through all possible values, when values
777:for the type of magic square we commonly call
577:magic. Because of the confusion with the term
566:Universal Classification System for Hypercubes
3421:
8:
1614:
1599:
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1561:
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1519:
1513:
1501:
1105:
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1066:
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947:
923:
560:is called the order of the magic hypercube.
3388:http://www.magichypercubes.com/Encyclopedia
1971:http://www.magichypercubes.com/Encyclopedia
1729:This positions the number 'k' at position:
362:of the hypercube, and is sometimes denoted
61:Learn how and when to remove these messages
3428:
3414:
3406:
3253:this is a n-dimensional version of (pe.):
3181:On the General Properties of Nasik Squares
2865:Σ ((reflect(k)) ? 2 : 0) ;
2432:expresses and all its r-agonal neighbors.
872:pandiagonal magic squares parallel to the
793:. In another 1878 paper he showed another
467:, with sequence of magic numbers given by
1973:at several places (especially p-section)
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299:Learn how and when to remove this message
281:Learn how and when to remove this message
223:Learn how and when to remove this message
158:Learn how and when to remove this message
3194:On the General Properties of Nasik Cubes
3166:Frost, A. H., Invention of Magic Cubes,
2258:0 ; θ ε {-1,1} > ; p,q ε
864:correctly summing lines. They also had 3
198:need to be rewritten in AMS-LaTeX markup
3327:Magic Cubes and Hypercubes - References
3159:
2900: : monagonal permutation (axis ε )
2132: : monagonal permutation (axis ε )
1620:{\displaystyle \langle {}_{k}i\rangle }
805:. He referred to all of these cubes as
463:= 4, a magic hypercube may be called a
3267:Alan Adler magic square multiplication
3265:this is a hyperbeam version of (pe.):
3255:Alan Adler magic square multiplication
1627:). The method starts at the position P
681:Or, to put it more concisely, all pan-
544:If, in addition, the numbers on every
499:-dimensional magic hypercube of order
2399:} might be put in notation as :
1592:) to more general movements (vectors
484:of the magic hypercube is called its
7:
3363:More on this alternative definition.
2537: : the amount of numbers along
1948:'s are combined into the hypercube:
621:numbers passing through each of the
96:adding citations to reliable sources
3333:An algorithm for making magic cubes
801:lines sum correctly i.e. Hendricks
753:pandiagonal magic squares of order
3205:Heinz, H.D., and Hendricks, J.R.,
2483:, a series that mimics the series
1820:
1385:
1367:
1303:
797:magic cube and a cube where all 13
789:, and an order-8 cube we class as
25:
3207:Magic Square Lexicon: Illustrated
2826:Only one direction with order = 2
2738:all orders are either even or odd
2331:} : {pan r-agonal; r = 1..n}
1932:of radix m numbers (also called "
749:cells. (This cube also contains 9
42:This article has multiple issues.
3484:
3209:, 2000, 0-9687985-0-0 pp 119-122
3168:Quarterly Journal of Mathematics
2568:: positions within the hyperbeam
2475:generalises the two dimensional
1004:: positions within the hypercube
678:the dimension of the hypercube.
587:is now the preferred term for
239:
174:
72:
31:
3398:Mitsutoshi Nakamura: Rectangles
2785:This is with the exception of m
2774:turns out integer is when all m
2576:: vectors through the hyperbeam
2372:some minor qualifications are:
1902:Latin prescription construction
1845:
83:needs additional citations for
50:or discuss these issues on the
3240:Rosser, B. and Walker, R. J.,
2778:are even. Thus suffices: all m
2503:It is customary to denote the
1829:
1814:
1795:
1792:
1713:
1668:
1123:: vector through the hypercube
436:
417:
405:
399:
1:
3526:Prime reciprocal magic square
3393:Marián Trenklar Cube-Ref.html
3329:Collected by Marian Trenkler
2945:-2 (by monagonal permutation)
2469:n-dimensional magic rectangle
2120: : component permutation
196:the mathematical expressions
3218:Planck, C., M.A., M.R.C.S.,
2507:with the letter 'n' and the
2414:} can simply be written as:
2324:} : {1-agonal n-agonal}
552:; otherwise, it is called a
3343:Articles by Christian Boyer
2357:gives arguments for using {
2349:Trump/Boyer {diagonal} cube
1411:{\displaystyle \left=\left}
1247:{\displaystyle \left=\left}
685:-agonals sum correctly for
554:semiperfect magic hypercube
379:, then it has magic number
194:. The specific problem is:
3673:
3323:Articles by Aale de Winkel
3196:, QJM, 15, 1878, pp 93-123
2925:permutation of m numbers.
2479:and the three dimensional
2237:i> ; i ε {-1,0,1}
1890:gives in his 1905 article
537:
3540:Higher dimensional shapes
3531:Most-perfect magic square
3482:
3220:The Theory of Paths Nasik
3183:, QJM, 15, 1878, pp 34-49
2449:most-perfect magic square
1894:The theory of Path Nasiks
1478:of the n numbers 0..n-1.
3585:Pandiagonal magic square
3580:Associative magic square
3521:Pandiagonal magic square
2782:are either even or odd.
2766:When any of the orders m
2233:i + 1) 3 <==> <
731:pandiagonal magic square
534:Perfect magic hypercubes
511:is different from 2 or
1490:KnightJump construction
670:is the magic constant,
617:(3 − 1) lines of
550:perfect magic hypercube
3349:A magic cube generator
3321:The Magic Encyclopedia
3046:The "normal hyperbeam"
2908:Coördinate permutation
2154:Coordinate permutation
1878:
1791:
1720:
1652:
1621:
1586:
1451:
1412:
1248:
1154:There can be multiple
1117:
998:
858:
844:
629:Nasik magic hypercubes
597:possible lines sum to
591:magic hypercube where
520:R programming language
503:exists if and only if
449:
261:by rewriting it in an
2997: ; i = 0..n-1) (
2941:< ; i = 0..m
2920:Monagonal permutation
2549: − 1.
2355:Nasik magic hypercube
2178:is the special case:
2166:Monagonal permutation
2146:Component permutation
1879:
1765:
1721:
1653:
1651:{\displaystyle V_{0}}
1622:
1587:
1452:
1450:{\displaystyle \left}
1418:("axial"-neighbor of
1413:
1249:
1118:
999:
997:{\displaystyle \left}
636:Nasik magic hypercube
450:
3287:Discrete Mathematics
2685:(m..) abbreviates: m
2614:basic multiplication
2445:Kathleen Ollerenshaw
2361:} as synonymous to {
2344:and divide it by m.
2186:n-agonal permutation
2138: : digit change
1987:basic multiplication
1736:
1665:
1635:
1596:
1498:
1422:
1277:
1164:
1010:
891:
386:
205:improve this article
190:to meet Knowledge's
92:improve this article
3621:Eight queens puzzle
3087:"Dynamic numbering"
3079:"Dynamic numbering"
2883:Basic manipulations
2342:Faulhaber's formula
2109:Basic manipulations
522:includes a module,
3347:magichypercube.com
3335:by Marian Trenkler
3290:207 (1999), 65-72.
3170:, 7,1866, pp92-102
3141:Perfect magic cube
3131:Magic cube classes
3081:makes use of the
3038:Special hyperbeams
2842:A hyperbeam knows
2071:A hypercube knows
1874:
1716:
1648:
1617:
1582:
1447:
1408:
1244:
1113:
994:
765:cells, and so on.
540:Magic Cube Classes
445:
326:generalization of
263:encyclopedic style
250:is written like a
3644:
3643:
3590:Multimagic square
3501:Alphamagic square
3148:John R. Hendricks
2935:"normal position"
2875:just as with the
2299:= m (m - 1) / 2.
2097:"normal position"
1825:
1819:
1384:
1366:
1360:
1345:
1342:
1302:
1074:
1035:
955:
916:
530:a multiple of 4.
443:
309:
308:
301:
291:
290:
283:
233:
232:
225:
192:quality standards
183:This article may
168:
167:
160:
142:
107:"Magic hypercube"
65:
16:(Redirected from
3664:
3599:Related concepts
3506:Antimagic square
3488:
3430:
3423:
3416:
3407:
3269:
3263:
3257:
3251:
3245:
3238:
3232:
3229:
3223:
3216:
3210:
3203:
3197:
3190:
3184:
3177:
3171:
3164:
3101:The "constant 1"
3091:magic hypercubes
2956:magic hypercubes
2877:magic hypercubes
2574:
2573:⟨⟩
2493:magic hypercubes
2441:compact complete
1883:
1881:
1880:
1875:
1867:
1866:
1841:
1840:
1823:
1817:
1813:
1812:
1790:
1779:
1761:
1760:
1748:
1747:
1725:
1723:
1722:
1719:{\displaystyle }
1717:
1712:
1711:
1693:
1692:
1680:
1679:
1657:
1655:
1654:
1649:
1647:
1646:
1626:
1624:
1623:
1618:
1610:
1609:
1604:
1591:
1589:
1588:
1583:
1456:
1454:
1453:
1448:
1446:
1442:
1438:
1437:
1432:
1417:
1415:
1414:
1409:
1407:
1403:
1382:
1364:
1358:
1354:
1353:
1348:
1343:
1340:
1336:
1335:
1330:
1319:
1315:
1300:
1293:
1292:
1287:
1264:is referred to.
1256:Of course given
1253:
1251:
1250:
1245:
1243:
1239:
1235:
1234:
1229:
1220:
1219:
1214:
1203:
1199:
1195:
1194:
1189:
1180:
1179:
1174:
1135:As is indicated
1122:
1120:
1119:
1114:
1112:
1108:
1072:
1033:
1026:
1025:
1020:
1003:
1001:
1000:
995:
993:
989:
953:
914:
907:
906:
901:
856:
842:
733:then would be a
725:
723:
722:
719:
716:
665:
663:
662:
659:
656:
616:
614:
613:
610:
607:
525:
480:The side-length
476:
454:
452:
451:
446:
444:
439:
429:
428:
412:
398:
397:
304:
297:
286:
279:
275:
272:
266:
243:
242:
235:
228:
221:
217:
214:
208:
178:
177:
170:
163:
156:
152:
149:
143:
141:
100:
76:
68:
57:
35:
34:
27:
21:
3672:
3671:
3667:
3666:
3665:
3663:
3662:
3661:
3647:
3646:
3645:
3640:
3616:Number Scrabble
3594:
3568:
3564:Magic hyperbeam
3559:Magic hypercube
3535:
3511:Geomagic square
3489:
3480:
3439:
3434:
3317:
3278:
3276:Further reading
3273:
3272:
3264:
3260:
3252:
3248:
3239:
3235:
3230:
3226:
3217:
3213:
3204:
3200:
3191:
3187:
3178:
3174:
3165:
3161:
3156:
3127:
3116:
3115:
3111:
3103:
3096:
3073:
3069:
3065:
3061:
3060:
3056:
3048:
3040:
3032:
3028:
3024:
3020:
3016:
3004:
3000:
2996:
2977:
2973:
2969:
2965:
2952:
2944:
2931:
2929:normal position
2922:
2910:
2885:
2866:
2864:
2860:
2858:
2854:
2840:
2833:
2828:
2818:
2814:
2810:
2804:
2800:
2796:
2788:
2781:
2777:
2773:
2769:
2761:
2757:
2753:
2749:
2740:
2735:
2728:
2727:
2721:
2720:
2714:
2713:
2707:
2706:
2700:
2696:
2692:
2688:
2681:
2680:
2676:
2670:
2669:
2663:
2662:
2656:
2652:
2650:
2644:
2643:
2639:
2633:
2632:
2626:
2625:
2610:
2601:
2596:
2590:
2589:
2585:
2572:
2559:
2534:
2501:
2477:magic rectangle
2473:magic hyperbeam
2465:magic hyperbeam
2461:
2459:Magic hyperbeam
2428:
2424:
2407:
2403:
2383:
2300:
2298:
2291:
2286:
2281:
2279:
2275:
2271:
2267:
2259:
2257:
2253:
2249:
2238:
2236:
2232:
2228:
2224:
2212:
2197:
2192:
2182:
2168:
2156:
2148:
2111:
2103:
2089:
2087:
2083:
2076:
2069:
2061:
2060:
2053:
2052:
2051:
2047:
2041:
2040:
2034:
2033:
2027:
2023:
2022:
2016:
2015:
2011:
2005:
2004:
1998:
1997:
1983:
1964:
1962:
1958:
1954:
1947:
1939:
1936:"). On these LP
1930:
1928:
1924:
1920:
1916:
1912:
1904:
1858:
1832:
1804:
1752:
1739:
1734:
1733:
1697:
1684:
1671:
1663:
1662:
1638:
1633:
1632:
1630:
1602:
1594:
1593:
1496:
1495:
1492:
1484:
1430:
1429:
1425:
1420:
1419:
1346:
1328:
1327:
1323:
1285:
1284:
1280:
1275:
1274:
1260:also one value
1227:
1212:
1211:
1207:
1187:
1172:
1171:
1167:
1162:
1161:
1132:
1018:
1017:
1013:
1008:
1007:
899:
898:
894:
889:
888:
882:
857:
854:
843:
840:
815:who hails from
781:and often call
771:
720:
717:
714:
713:
711:
660:
657:
647:
646:
644:
631:
611:
608:
605:
604:
602:
542:
536:
523:
490:J. R. Hendricks
468:
465:magic tesseract
420:
413:
389:
384:
383:
370:
356:space diagonals
317:magic hypercube
305:
294:
293:
292:
287:
276:
270:
267:
259:help improve it
256:
244:
240:
229:
218:
212:
209:
202:
179:
175:
164:
153:
147:
144:
101:
99:
89:
77:
36:
32:
23:
22:
18:Magic tesseract
15:
12:
11:
5:
3670:
3668:
3660:
3659:
3649:
3648:
3642:
3641:
3639:
3638:
3633:
3628:
3626:Magic constant
3623:
3618:
3613:
3608:
3602:
3600:
3596:
3595:
3593:
3592:
3587:
3582:
3576:
3574:
3573:Classification
3570:
3569:
3567:
3566:
3561:
3556:
3555:
3554:
3543:
3541:
3537:
3536:
3534:
3533:
3528:
3523:
3518:
3513:
3508:
3503:
3497:
3495:
3494:Related shapes
3491:
3490:
3483:
3481:
3479:
3478:
3476:Magic triangle
3473:
3468:
3463:
3461:Magic hexagram
3458:
3453:
3447:
3445:
3441:
3440:
3437:Magic polygons
3435:
3433:
3432:
3425:
3418:
3410:
3404:
3403:
3400:
3395:
3390:
3385:
3380:
3375:
3370:
3365:
3360:
3355:
3350:
3344:
3341:multimagie.com
3338:
3337:
3336:
3324:
3316:
3315:External links
3313:
3312:
3311:
3308:
3305:
3302:
3291:
3282:
3277:
3274:
3271:
3270:
3258:
3246:
3233:
3224:
3211:
3198:
3185:
3179:Frost, A. H.,
3172:
3158:
3157:
3155:
3152:
3151:
3150:
3145:
3144:
3143:
3133:
3126:
3123:
3119:
3118:
3113:
3109:
3107:
3102:
3099:
3094:
3075:
3074:
3071:
3067:
3063:
3058:
3054:
3052:
3047:
3044:
3039:
3036:
3035:
3034:
3030:
3026:
3022:
3018:
3014:
3007:
3006:
3002:
2998:
2994:
2980:
2979:
2975:
2971:
2967:
2963:
2951:
2948:
2947:
2946:
2942:
2930:
2927:
2921:
2918:
2909:
2906:
2902:
2901:
2895:
2884:
2881:
2862:
2856:
2852:
2850:
2848:
2839:
2836:
2831:
2827:
2824:
2820:
2819:
2816:
2812:
2808:
2805:
2802:
2798:
2794:
2786:
2779:
2775:
2771:
2767:
2764:
2763:
2759:
2755:
2751:
2747:
2739:
2736:
2734:
2731:
2725:
2723:
2718:
2716:
2711:
2709:
2704:
2702:
2701:abbreviates: m
2698:
2694:
2690:
2686:
2683:
2682:
2678:
2674:
2672:
2667:
2665:
2660:
2658:
2654:
2648:
2646:
2641:
2637:
2635:
2630:
2628:
2623:
2621:
2609:
2608:Multiplication
2606:
2600:
2597:
2595:
2592:
2587:
2583:
2581:
2578:
2577:
2569:
2558:
2555:
2551:
2550:
2530:
2523:
2500:
2497:
2460:
2457:
2439:for squares: {
2434:
2433:
2430:
2426:
2422:
2405:
2401:
2393:
2392:
2385:
2381:
2333:
2332:
2325:
2318:
2311:
2296:
2294:
2289:
2285:
2284:Qualifications
2282:
2277:
2273:
2269:
2265:
2263:
2255:
2251:
2247:
2245:
2234:
2230:
2226:
2222:
2220:
2211:
2208:
2196:
2193:
2190:
2180:
2174:Noted be that
2167:
2164:
2155:
2152:
2147:
2144:
2140:
2139:
2133:
2127:
2121:
2110:
2107:
2101:
2085:
2081:
2079:
2068:
2065:
2058:
2056:
2049:
2045:
2043:
2038:
2036:
2031:
2029:
2025:
2020:
2018:
2013:
2009:
2007:
2002:
2000:
1995:
1993:
1991:
1982:
1981:Multiplication
1979:
1960:
1956:
1952:
1950:
1945:
1942:digit changing
1937:
1926:
1922:
1918:
1914:
1910:
1908:
1903:
1900:
1885:
1884:
1873:
1870:
1865:
1861:
1857:
1854:
1851:
1848:
1844:
1839:
1835:
1831:
1828:
1822:
1816:
1811:
1807:
1803:
1800:
1797:
1794:
1789:
1786:
1783:
1778:
1775:
1772:
1768:
1764:
1759:
1755:
1751:
1746:
1742:
1727:
1726:
1715:
1710:
1707:
1704:
1700:
1696:
1691:
1687:
1683:
1678:
1674:
1670:
1645:
1641:
1628:
1616:
1613:
1608:
1601:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1491:
1488:
1483:
1480:
1474:" specifies a
1445:
1441:
1436:
1428:
1406:
1402:
1399:
1396:
1393:
1390:
1387:
1381:
1378:
1375:
1372:
1369:
1363:
1357:
1352:
1339:
1334:
1326:
1322:
1318:
1314:
1311:
1308:
1305:
1299:
1296:
1291:
1283:
1242:
1238:
1233:
1226:
1223:
1218:
1210:
1206:
1202:
1198:
1193:
1186:
1183:
1178:
1170:
1128:
1125:
1124:
1111:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1032:
1029:
1024:
1016:
1005:
992:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
919:
913:
910:
905:
897:
881:
878:
874:space-diagonal
852:
838:
770:
767:
674:the order and
630:
627:
535:
532:
524:library(magic)
457:
456:
442:
438:
435:
432:
427:
423:
419:
416:
410:
407:
404:
401:
396:
392:
366:
360:magic constant
334:, that is, an
307:
306:
289:
288:
247:
245:
238:
231:
230:
182:
180:
173:
166:
165:
80:
78:
71:
66:
40:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3669:
3658:
3657:Magic squares
3655:
3654:
3652:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3604:
3603:
3601:
3597:
3591:
3588:
3586:
3583:
3581:
3578:
3577:
3575:
3571:
3565:
3562:
3560:
3557:
3553:
3550:
3549:
3548:
3545:
3544:
3542:
3538:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3498:
3496:
3492:
3487:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3456:Magic hexagon
3454:
3452:
3449:
3448:
3446:
3442:
3438:
3431:
3426:
3424:
3419:
3417:
3412:
3411:
3408:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3348:
3345:
3342:
3339:
3334:
3331:
3330:
3328:
3325:
3322:
3319:
3318:
3314:
3309:
3306:
3303:
3300:
3299:0-9687985-0-0
3296:
3292:
3289:
3288:
3283:
3280:
3279:
3275:
3268:
3262:
3259:
3256:
3250:
3247:
3243:
3237:
3234:
3228:
3225:
3221:
3215:
3212:
3208:
3202:
3199:
3195:
3192:Frost, A. H.
3189:
3186:
3182:
3176:
3173:
3169:
3163:
3160:
3153:
3149:
3146:
3142:
3139:
3138:
3137:
3134:
3132:
3129:
3128:
3124:
3122:
3105:
3104:
3100:
3098:
3092:
3088:
3084:
3080:
3050:
3049:
3045:
3043:
3037:
3012:
3011:
3010:
2992:
2991:
2990:
2987:
2985:
2961:
2960:
2959:
2957:
2950:Qualification
2949:
2940:
2939:
2938:
2936:
2928:
2926:
2919:
2917:
2915:
2907:
2905:
2899:
2896:
2893:
2890:
2889:
2888:
2882:
2880:
2878:
2874:
2869:
2847:
2845:
2837:
2835:
2825:
2823:
2806:
2792:
2791:
2790:
2783:
2745:
2744:
2743:
2737:
2732:
2730:
2619:
2618:
2617:
2616:is given by:
2615:
2607:
2605:
2598:
2593:
2591:
2575:
2570:
2567:
2564:
2563:
2562:
2556:
2554:
2548:
2544:
2541:th monagonal
2540:
2536:
2533:
2529:
2524:
2521:
2519:
2514:
2513:
2512:
2510:
2506:
2498:
2496:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2458:
2456:
2452:
2450:
2446:
2442:
2437:
2431:
2421:
2420:
2419:
2417:
2413:
2409:
2398:
2390:
2386:
2379:
2375:
2374:
2373:
2370:
2368:
2364:
2360:
2356:
2352:
2350:
2345:
2343:
2337:
2330:
2326:
2323:
2319:
2316:
2312:
2309:
2305:
2304:
2303:
2293:
2288:A hypercube H
2283:
2262:
2244:
2241:
2219:
2217:
2209:
2207:
2205:
2201:
2195:Digitchanging
2194:
2189:
2187:
2179:
2177:
2172:
2165:
2163:
2161:
2153:
2151:
2145:
2143:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2115:
2114:
2108:
2106:
2100:
2098:
2093:
2078:
2074:
2066:
2064:
1990:
1989:is given by:
1988:
1980:
1978:
1974:
1972:
1968:
1967:J.R.Hendricks
1949:
1943:
1935:
1907:
1901:
1899:
1897:
1895:
1889:
1871:
1868:
1863:
1859:
1855:
1852:
1849:
1846:
1842:
1837:
1833:
1826:
1809:
1805:
1798:
1787:
1784:
1781:
1776:
1773:
1770:
1766:
1762:
1757:
1753:
1749:
1744:
1740:
1732:
1731:
1730:
1708:
1705:
1702:
1698:
1694:
1689:
1685:
1681:
1676:
1672:
1661:
1660:
1659:
1643:
1639:
1611:
1606:
1576:
1573:
1570:
1567:
1564:
1558:
1552:
1549:
1546:
1543:
1537:
1531:
1528:
1525:
1522:
1516:
1510:
1507:
1504:
1489:
1487:
1481:
1479:
1477:
1473:
1468:
1465:
1461:
1458:
1443:
1439:
1434:
1426:
1404:
1400:
1397:
1394:
1391:
1388:
1379:
1376:
1373:
1370:
1361:
1355:
1350:
1337:
1332:
1324:
1320:
1316:
1312:
1309:
1306:
1297:
1294:
1289:
1281:
1272:
1270:
1265:
1263:
1259:
1254:
1240:
1236:
1231:
1224:
1221:
1216:
1208:
1204:
1200:
1196:
1191:
1184:
1181:
1176:
1168:
1159:
1157:
1152:
1150:
1146:
1142:
1138:
1133:
1131:
1109:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1078:
1075:
1069:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1039:
1036:
1030:
1027:
1022:
1014:
1006:
990:
983:
980:
977:
974:
971:
968:
965:
959:
956:
950:
944:
941:
938:
935:
932:
929:
926:
920:
917:
911:
908:
903:
895:
887:
886:
885:
879:
877:
875:
871:
867:
863:
851:
847:
837:
832:
830:
826:
822:
818:
814:
810:
809:
804:
800:
796:
792:
788:
784:
780:
776:
768:
766:
764:
760:
756:
752:
748:
744:
740:
736:
732:
727:
709:
705:
700:
698:
697:
692:
688:
684:
679:
677:
673:
669:
654:
650:
642:
638:
637:
628:
626:
624:
620:
600:
596:
595:
590:
586:
585:
580:
576:
572:
568:
567:
561:
559:
556:. The number
555:
551:
547:
546:cross section
541:
533:
531:
529:
521:
516:
514:
510:
506:
502:
498:
493:
491:
487:
483:
478:
475:
471:
466:
462:
440:
433:
430:
425:
421:
414:
408:
402:
394:
390:
382:
381:
380:
378:
374:
369:
365:
361:
357:
353:
349:
345:
341:
337:
333:
329:
328:magic squares
325:
323:
318:
314:
303:
300:
285:
282:
274:
264:
260:
254:
253:
248:This article
246:
237:
236:
227:
224:
216:
206:
201:
199:
193:
189:
188:
181:
172:
171:
162:
159:
151:
140:
137:
133:
130:
126:
123:
119:
116:
112:
109: –
108:
104:
103:Find sources:
97:
93:
87:
86:
81:This article
79:
75:
70:
69:
64:
62:
55:
54:
49:
48:
43:
38:
29:
28:
19:
3636:Magic series
3606:Latin square
3558:
3516:Heterosquare
3466:Magic square
3451:Magic circle
3285:
3261:
3249:
3241:
3236:
3227:
3219:
3214:
3206:
3201:
3193:
3188:
3180:
3175:
3167:
3162:
3120:
3117: : = 1
3076:
3041:
3008:
2988:
2983:
2981:
2953:
2934:
2932:
2923:
2913:
2911:
2903:
2897:
2891:
2886:
2872:
2870:
2867:
2861: ; R =
2843:
2841:
2829:
2821:
2784:
2765:
2741:
2684:
2613:
2611:
2602:
2594:Construction
2579:
2571:
2565:
2560:
2552:
2546:
2542:
2538:
2531:
2527:
2525:
2517:
2515:
2502:
2485:magic square
2480:
2476:
2472:
2468:
2464:
2462:
2453:
2440:
2438:
2435:
2415:
2411:
2400:
2396:
2394:
2388:
2377:
2371:
2366:
2362:
2358:
2353:
2346:
2338:
2334:
2328:
2321:
2315:pan r-agonal
2314:
2307:
2301:
2287:
2260:
2242:
2239:
2215:
2213:
2203:
2200:
2198:
2185:
2183:
2175:
2173:
2169:
2159:
2157:
2149:
2141:
2135:
2129:
2123:
2117:
2112:
2104:
2096:
2094:
2090:
2072:
2070:
2054:
1986:
1984:
1975:
1966:
1965:
1941:
1933:
1931:
1905:
1893:
1887:
1886:
1728:
1493:
1485:
1482:Construction
1472:perm(0..n-1)
1471:
1469:
1466:
1462:
1459:
1273:
1268:
1266:
1261:
1257:
1255:
1160:
1155:
1153:
1148:
1144:
1140:
1136:
1134:
1129:
1126:
883:
869:
865:
861:
859:
849:
845:
834:
823:District in
813:D R Kaprekar
807:
806:
802:
798:
794:
791:pantriagonal
790:
786:
782:
778:
774:
772:
762:
758:
754:
750:
746:
742:
738:
734:
728:
707:
703:
701:
695:
694:
690:
686:
682:
680:
675:
671:
667:
652:
648:
640:
635:
634:
632:
622:
618:
598:
593:
592:
588:
583:
582:
578:
574:
570:
565:
564:
562:
557:
553:
549:
543:
527:
517:
512:
508:
507:> 1 and
504:
500:
496:
494:
485:
481:
479:
464:
460:
458:
376:
372:
367:
363:
347:
343:
339:
335:
324:-dimensional
321:
316:
310:
295:
277:
271:October 2017
268:
249:
219:
210:
203:Please help
195:
184:
154:
148:October 2010
145:
135:
128:
121:
114:
102:
90:Please help
85:verification
82:
58:
51:
45:
44:Please help
41:
3631:Magic graph
3611:Word square
3083:isomorphism
3062: : =
2733:Curiosities
2520:) Dimension
2499:Conventions
2225:where: p =
2216:pathfinders
2210:Pathfinders
2191:_ = _(2-1)
1929:) % m
1476:permutation
1271:= 1 as in:
825:Maharashtra
795:pandiagonal
787:pandiagonal
779:pandiagonal
332:magic cubes
313:mathematics
207:if you can.
3547:Magic cube
3471:Magic star
3402:Peace Cube
3154:References
3136:Magic cube
2545:= 0, ...,
2489:magic cube
2481:magic beam
2416:+ = m - 1
2176:reflection
1470:Further: "
538:See also:
213:March 2014
118:newspapers
47:improve it
3093:of order
2993:S = lcm(m
2914:transpose
2557:Notations
2505:dimension
2160:transpose
1888:C. Planck
1869:−
1856:…
1821:%
1802:∖
1785:−
1767:∑
1706:−
1695:…
1615:⟩
1600:⟨
1580:⟩
1574:−
1565:−
1562:⟨
1556:⟩
1544:−
1541:⟨
1535:⟩
1529:−
1520:⟨
1514:⟩
1502:⟨
1398:−
1386:#
1368:#
1304:#
1100:−
1091:⋯
1079:∈
1061:−
1052:⋯
1040:∈
981:−
972:⋯
960:∈
942:−
933:⋯
921:∈
880:Notations
702:The term
350:array of
53:talk page
3651:Category
3125:See also
3033:- 1) / 2
3005:- 1) / 2
2978:- 1) / 2
2762:- 1) / 2
2634: :
2412:complete
2404:Σ = 2 S
2389:complete
2308:r-agonal
2181:~R = _R
2006: :
1110:⟩
1015:⟨
876:planes.
853:—
839:—
352:integers
346:× ... ×
185:require
3552:classes
2838:Aspects
2693:. (m..)
2535:) Order
2447:called
2418:where:
2397:compact
2378:compact
2367:perfect
2363:perfect
2329:perfect
2280:0 >
2067:Aspects
817:Deolali
803:perfect
783:perfect
769:History
724:
712:
696:perfect
664:
645:
625:cells.
615:
603:
579:perfect
575:perfect
474:A021003
472::
319:is the
257:Please
187:cleanup
132:scholar
3297:
2566:; i= ]
2509:orders
2084:; R =
1934:digits
1824:
1818:
1383:
1365:
1359:
1344:
1341:
1301:
1073:
1034:
954:
915:
836:paper.
689:= 1...
666:where
134:
127:
120:
113:
105:
3444:Types
3112:,..,m
3097:Π m.
3057:,..,m
2984:magic
2834:= 2.
2715:,..,m
2697:(m..)
2689:,..,m
2677:(m..)
2673:(m..)
2666:(m..)
2659:(m..)
2647:(m..)
2640:(m..)
2636:(m..)
2629:(m..)
2622:(m..)
2599:Basic
2586:,..,m
2359:nasik
2322:magic
2264:<
2246:<
2204:perm(
829:India
821:Nasik
808:nasik
775:Nasik
759:nasik
757:.) A
743:nasik
735:nasik
715:3 − 1
704:nasik
584:nasik
486:order
139:JSTOR
125:books
3295:ISBN
2937:by:
2873:n! 2
2384:/ m}
2099:by:
2073:n! 2
1959:Σ LP
1940:'s "
1925:+ LP
1917:Σ LP
1913:= (
518:The
470:OEIS
459:For
330:and
315:, a
111:news
3114:n-1
3095:k=0
3089:of
3070:i m
3064:k=0
3059:n-1
3027:j=0
3019:j=0
3015:max
2999:j=0
2972:j=0
2966:= m
2863:k=0
2857:n-1
2855:..m
2817:m,1
2815:* N
2813:1,m
2811:= N
2803:1,m
2801:* N
2799:m,1
2797:= N
2756:j=0
2750:= m
2724:n-1
2717:n-1
2691:n-1
2651:k=0
2645:= ]
2627:* B
2588:n-1
2423:(k)
2410:. {
2408:/ m
2402:(k)
2276:-1
2229:Σ (
2227:k=0
2202:in
2086:k=0
1999:* H
1957:k=0
1927:k,n
1919:k,l
1915:l=0
819:in
594:all
589:any
571:all
311:In
94:by
3653::
3066:Σ
3029:Πm
3021:Πm
3017:=
3001:Πm
2986:}
2974:Πm
2898:_2
2851:(m
2758:Πm
2729:.
2664:+
2655:k1
2653:Πm
2487:,
2463:A
2429:1.
2272:1
2268:1
2254:θ
2250:1
2221:Pf
2130:_2
2035:+
2017:=
1963:m
1955:=
1909:LP
1872:1.
1457:)
827:,
729:A
726:.
710:=
655:+1
643:=
633:A
581:,
492:.
477:.
342:×
338:×
56:.
3429:e
3422:t
3415:v
3301:.
3110:0
3108:m
3106:1
3072:k
3068:k
3055:0
3053:m
3051:N
3031:j
3025:(
3023:j
3013:S
3003:j
2995:i
2976:j
2970:(
2968:k
2964:k
2962:S
2943:k
2892:^
2859:)
2853:0
2849:B
2844:2
2832:k
2809:m
2807:N
2795:m
2793:N
2787:k
2780:k
2776:k
2772:k
2768:k
2760:j
2754:(
2752:k
2748:k
2746:S
2726:2
2722:m
2719:1
2712:2
2710:0
2708:m
2705:1
2703:0
2699:2
2695:1
2687:0
2679:2
2675:1
2671:]
2668:2
2661:2
2657:]
2649:1
2642:2
2638:1
2631:2
2624:1
2620:B
2584:0
2582:m
2547:n
2543:k
2539:k
2532:k
2528:m
2526:(
2518:n
2516:(
2467:(
2427:k
2406:m
2395:{
2387:{
2382:m
2376:{
2327:{
2320:{
2313:{
2306:{
2297:m
2295:S
2290:m
2278:s
2274:l
2270:k
2266:j
2256:l
2252:k
2248:j
2235:k
2231:k
2223:p
2136:=
2124:^
2118:#
2082:m
2080:H
2059:2
2057:m
2050:2
2048:m
2046:1
2044:m
2042:]
2039:2
2037:m
2032:2
2030:m
2028:]
2026:1
2024:m
2021:1
2019:m
2014:2
2012:m
2010:1
2008:m
2003:2
2001:m
1996:1
1994:m
1992:H
1961:k
1953:m
1951:H
1946:k
1938:k
1923:l
1921:x
1911:k
1896:"
1892:"
1864:n
1860:m
1853:0
1850:=
1847:k
1843:;
1838:l
1834:V
1830:)
1827:m
1815:)
1810:l
1806:m
1799:k
1796:(
1793:(
1788:1
1782:n
1777:0
1774:=
1771:l
1763:+
1758:0
1754:P
1750:=
1745:k
1741:P
1714:]
1709:1
1703:n
1699:V
1690:0
1686:V
1682:,
1677:0
1673:P
1669:[
1644:0
1640:V
1629:0
1612:i
1607:k
1577:2
1571:,
1568:1
1559:,
1553:2
1550:,
1547:1
1538:,
1532:2
1526:,
1523:1
1517:,
1511:2
1508:,
1505:1
1444:]
1440:0
1435:k
1427:[
1405:]
1401:1
1395:n
1392:=
1389:j
1380:;
1377:1
1374:=
1371:k
1362:;
1356:0
1351:j
1338:1
1333:k
1325:[
1321:=
1317:]
1313:1
1310:=
1307:k
1298:;
1295:1
1290:k
1282:[
1269:k
1262:i
1258:k
1241:]
1237:i
1232:1
1225:,
1222:j
1217:k
1209:[
1205:=
1201:]
1197:j
1192:k
1185:,
1182:i
1177:1
1169:[
1156:k
1149:m
1145:i
1141:i
1137:k
1130:m
1106:}
1103:1
1097:m
1094:,
1088:,
1085:0
1082:{
1076:i
1070:;
1067:}
1064:1
1058:n
1055:,
1049:,
1046:0
1043:{
1037:k
1031:;
1028:i
1023:k
991:]
987:}
984:1
978:m
975:,
969:,
966:0
963:{
957:i
951:;
948:}
945:1
939:n
936:,
930:,
927:0
924:{
918:k
912:;
909:i
904:k
896:[
870:m
866:m
862:m
799:m
763:m
755:m
751:m
747:m
739:m
721:2
718:/
708:P
691:n
687:r
683:r
676:n
672:m
668:S
661:2
658:/
653:m
651:(
649:m
641:S
623:m
619:m
612:2
609:/
606:1
599:S
558:n
528:n
513:p
509:n
505:p
501:n
497:p
482:n
461:k
455:.
441:2
437:)
434:1
431:+
426:k
422:n
418:(
415:n
409:=
406:)
403:n
400:(
395:k
391:M
377:n
373:n
371:(
368:k
364:M
348:n
344:n
340:n
336:n
322:k
302:)
296:(
284:)
278:(
273:)
269:(
265:.
226:)
220:(
215:)
211:(
200:.
161:)
155:(
150:)
146:(
136:·
129:·
122:·
115:·
88:.
63:)
59:(
20:)
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