3475:
63:
22:
230:
165:
2328:
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.
2159:
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.
2324:
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.
2443:
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
2913:
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
824:
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
2080:
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"!
1452:
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)
1871:
1110:
1579:
1965:
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)
2592:
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.
2051:
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.
1895:(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:
442:
1647:
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:
1614:
2250:
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:
3110:
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.
2044:
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
1724:
1405:
1241:
998:
2354:}. 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 {
1486:
849:
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
1116:
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:
2569:
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:
1645:
1444:
991:
839:
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).
240:
1713:
1256:
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. #
2139:
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
3074:
of every hyperbeam with this normal, changing the source, changes the hyperbeam. Basic multiplications of normal hyperbeams play a special role with the
2444:
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.
251:
2291:
Besides more specific qualifications the following are the most important, "summing" of course stands for "summing correctly to the magic sum"
695:
would apply to all dimensions of magic hypercubes in which the number of correctly summing paths (lines) through any cell of the hypercube is
2151:(usually denoted by ) is used with two dimensional matrices, in general though perhaps "coordinate permutation" might be preferable.
458:
2905:(usually denoted by ) is used with two dimensional matrices, in general though perhaps "coördinaatpermutation" might be preferable.
1456:
Note: as far as I know this notation is not in general use yet(?), Hypercubes are not generally analyzed in this particular manner.
3416:
2868:, directions with equal orders contribute factors depending on the hyperbeam's orders. This goes beyond the scope of this article.
3296:
Planck, C., M.A., M.R.C.S., The Theory of Paths Nasik, 1905, printed for private circulation. Introductory letter to the paper
127:
99:
2329:("1-magic" would be "monomagic" but "mono" is usually omitted). The sum for p-Multimagic hypercubes can be found by using
2229:
This gives 3 directions. since every direction is traversed both ways one can limit to the upper half of the full range.
2195:) since a component is filled with radix m digits, a permutation over m numbers is an appropriate manner to denote these.
820:. In 1905 Dr. Planck expanded on the nasik idea in his Theory of Paths Nasik. In the introductory to his paper, he wrote;
2901:
The exchange of coördinaat into , because of n coördinates a permutation over these n directions is required. The term
2147:
The exchange of coordinate into , because of n coordinates a permutation over these n directions is required. The term
106:
2601:
Amongst the various ways of compounding, the multiplication can be considered as the most basic of these methods. The
1974:
Amongst the various ways of compounding, the multiplication can be considered as the most basic of these methods. The
374:
3287:
287:
269:
211:
146:
49:
2094:(explicitly stated here: the minimum of all corner points. The axial neighbour sequentially based on axial number)
193:
80:
35:
1147:
between brackets, these cannot have the same value, though in undetermined order, which explains the equality of:
113:
2835:
Aspectial variants, which are obtained by coördinate reflection ( → ) effectively giving the
Aspectial variant:
2064:
Aspectial variants, which are obtained by coordinate reflection ( --> ) and coordinate permutations ( -->
3275:
3270:
Thomas R. Hagedorn, On the existence of magic n-dimensional rectangles, Discrete
Mathematics 207 (1999), 53-63.
1136:
are outside the range it is simply moved back into the range by adding or subtracting appropriate multiples of
84:
3514:
1584:
477:. Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by
552:
This definition of "perfect" assumes that one of the older definitions for perfect magic cubes is used. The
95:
3464:
3409:
628:
is a magic hypercube with the added restriction that all possible lines through each cell sum correctly to
2500:
of a hyperbeam with the letter 'm' (appended with the subscripted number of the direction it applies to).
2460:) is a variation on a magic hypercube where the orders along each direction may be different. As such a
3439:
3075:
3067:
2922:
In case no restrictions are considered on the n-agonals a magic hyperbeam can be represented shown in
1866:{\displaystyle P_{k}=P_{0}+\sum _{l=0}^{n-1}((k\backslash m^{l})\ \%\ m)V_{l};\quad k=0\dots m^{n}-1.}
3519:
3361:
2437:
2084:
Aspectial variants are generally seen as being equal. Thus any hypercube can be represented shown in
186:
180:
1105:{\displaystyle \left\langle {}_{k}i;\ k\in \{0,\cdots ,n-1\};\ i\in \{0,\cdots ,m-1\}\right\rangle }
3573:
3568:
3509:
3220:
Andrews, W. S., Magic
Squares and Cubes, Dover Publ. 1917. Essay pages 363-375 written by C. Planck
2860:
In case one views different orientations of the beam as equal one could view the number of aspects
2131:
Note: '#', '^', '_' and '=' are essential part of the notation and used as manipulation selectors.
1574:{\displaystyle \langle 1,2\rangle ,\langle 1,-2\rangle ,\langle -1,2\rangle ,\langle -1,-2\rangle }
719:
534:
508:
3282:
Harvey D. Heinz & John R. Hendricks, Magic Square
Lexicon: Illustrated, self-published, 2000,
244:
that states a
Knowledge editor's personal feelings or presents an original argument about a topic.
3402:
3255:
3243:
2857:
Where reflect(k) true if and only if coordinate k is being reflected, only then 2 is added to R.
2484:
article in close detail, and just as that article serves merely as an introduction to the topic.
1265:
1152:
73:
3645:
3499:
2330:
1880:
3386:
1958:
often uses modular equation, conditions to make hypercubes of various quality can be found on
3351:
2343:
120:
2433:
1623:
1410:
879:
41:
3293:
J.R.Hendricks: Magic
Squares to Tesseract by Computer, Self-published, 1998, 0-9684700-0-9
8:
3609:
3356:
3309:
2893:
Note: '^' and '_' are essential part of the notation and used as manipulation selectors.
1475:
Besides more specific constructions two more general construction method are noticeable:
343:
such that the sums of the numbers on each pillar (along any axis) as well as on the main
247:
3540:
3129:
3119:
2337:
528:
1653:
3578:
3489:
3283:
3136:
1887:
conditions to create with this method "Path Nasik" (or modern {perfect}) hypercubes.
590:. Nasik was defined in this manner by C. Planck in 1905. A nasik magic hypercube has
175:
3494:
3341:
3079:
2944:
2865:
2481:
2232:
With these pathfinders any line to be summed over (or r-agonal) can be specified:
3604:
3552:
2440:, {compact complete} is the qualifier for the feature in more than 2 dimensions.
478:
3366:
2978:
When the orders are not relatively prime the n-agonal sum can be restricted to:
537:
diagonal also sum up to the hypercube's magic number, the hypercube is called a
3614:
3449:
3342:
History, definitions, and examples of perfect magic cubes and other dimensions.
3321:
2077:Σ ((reflect(k)) ? 2 : 0) ; perm(0..n-1) a permutation of 0..n-1
862:
348:
344:
3376:
3299:
Marián
Trenkler, Magic rectangles, The Mathematical Gazette 83(1999), 102-105.
2876:
Besides more specific manipulations, the following are of more general nature
2102:
Besides more specific manipulations, the following are of more general nature
1959:
3639:
3444:
3425:
3233:, 1939. A bound volume at Cornell University, catalogued as QA 165 R82+pt.1-4
2819:
since any number has but one complement only one of the directions can have m
2091:= min() (by reflection) < ; k = 0..n-2 (by coordinate permutation)
1483:
This construction generalizes the movement of the chessboard horses (vectors
3381:
3371:
3315:
3066:
This hyperbeam can be seen as the source of all numbers. A procedure called
3624:
3594:
3504:
3454:
2473:
801:
316:
3367:
An ambitious ongoing work on classifications of magic cubes and tesseracts
3211:, 1905, printed for private circulation. Introductory letter to the paper.
2414:Σ is symbolic for summing all possible k's, there are 2 possibilities for
3619:
3599:
3071:
2497:
1464:
813:
774:. He then demonstrated the concept with an order-7 cube we now class as
301:
3347:
An alternative definition of
Perfect, with history of recent discoveries
2207:", these directions are simplest denoted in a ternary number system as:
1449:(#j=n-1 can be left unspecified) j now runs through all the values in .
3535:
3459:
3124:
2477:
320:
3346:
3474:
2542:
Further: In this article the analytical number range is being used.
2493:
2188:
Usually being applied at component level and can be seen as given by
309:
3372:
A variety of John R. Hendricks material, written under his direction
2173:
Further when all the axes undergo the same permutation (R = 2-1) an
62:
844:
W. S. Andrews, Magic
Squares and Cubes, Dover Publ., 1917, page 366
340:
2550:
in order to keep things in hand a special notation was developed:
2380:} : {all pairs halve an n-agonal apart sum equal (to (m - 1)}
1933:" (?i.e. Basic manipulation) are generally applied before these LP
873:
in order to keep things in hand a special notation was developed:
857:
pandiagonal magic squares parallel to the faces of the cube, and 6
504:= 1. A construction of a magic hypercube follows from the proof.
3394:
2177:
is achieved, In this special case the 'R' is usually omitted so:
1140:, as the magic hypercube resides in n-dimensional modular space.
805:
734:
magic cube would have 13 magic lines passing through each of its
3335:
2336:
Also "magic" (i.e. {1-agonal n-agonal}) is usually assumed, the
750:
magic tesseract would have 40 lines passing through each of its
562:
possible lines sum correctly for the hypercube to be considered
2281:
with numbers in the analytical numberrange has the magic sum:
558:(John R. Hendricks) requires that for any dimension hypercube,
3357:
A Magic Hypercube encyclopedia with a broad range of material
3329:
2943:
Qualifying the hyperbeam is less developed then it is on the
817:
809:
730:
cells. This was A.H. Frost’s original definition of nasik. A
682:. This definition is the same as the Hendricks definition of
2998:
with all orders relatively prime this reaches its maximum:
2203:
J. R. Hendricks called the directions within a hypercubes "
462:
364:). If a magic hypercube consists of the numbers 1, 2, ...,
241:
personal reflection, personal essay, or argumentative essay
2947:
in fact only the k'th monagonal direction need to sum to:
830:
C. Planck, M.A., M.R.C.S., The Theory of Paths Nasik, 1905
515:, that will create magic hypercubes of any dimension with
2811:
Which goes beyond the scope of this introductory article
2731:
A fact that can be easily seen since the magic sums are:
2432:} is the "modern/alternative qualification" of what Dame
2306:} : all (unbroken and broken) r-agonals are summing.
1620:
and further numbers are sequentially placed at positions
2778:=1 of course, which allows for general identities like:
2358:} means {pan r-agonal; r = 1..n} (as mentioned above).
2971:
for all k = 0..n-1 for the hyperbeam to be qualified {
1790:
2511: : the amount of directions within a hyperbeam.
1727:
1656:
1626:
1587:
1489:
1413:
1268:
1155:
1001:
882:
726:
square because 4 magic line pass through each of the
377:
2814:
2759:
is even, the product is even and thus the only way S
2726:
3034:
2425:for {complete} the complement of is at position .
2299:} : all main (unbroken) r-agonals are summing.
762:In 1866 and 1878, Rev. A. H. Frost coined the term
87:. Unsourced material may be challenged and removed.
2883: : coördinate permutation (n == 2: transpose)
2340:is technically seen {1-agonal 2-agonal 3-agonal}.
2115: : coordinate permutation (n == 2: transpose)
1890:
1865:
1707:
1639:
1608:
1573:
1438:
1399:
1235:
1128:runs through the dimensions, while the coordinate
1104:
985:
436:
3031:The following hyperbeams serve special purposes:
2480:and magic hypercube. This article will mimic the
835:In 1917, Dr. Planck wrote again on this subject.
688:, but different from the Boyer/Trump definition.
3637:
3273:Thomas R. Hagedorn, Magic rectangles revisited,
484:Marian Trenkler proved the following theorem: A
437:{\displaystyle M_{k}(n)={\frac {n(n^{k}+1)}{2}}}
347:are all the same. The common sum is called the
2369:} : {all order 2 subhyper cubes sum to 2 S
800:as a respect to the great Indian Mathematician
3362:A Unified classification system for hypercubes
3231:Magic Squares: Published papers and Supplement
2066:i]) effectively giving the Aspectial variant:
1132:runs through all possible values, when values
766:for the type of magic square we commonly call
566:magic. Because of the confusion with the term
555:Universal Classification System for Hypercubes
3410:
3089:
1603:
1588:
1568:
1550:
1544:
1529:
1523:
1508:
1502:
1490:
1094:
1070:
1055:
1031:
975:
951:
936:
912:
549:is called the order of the magic hypercube.
3377:http://www.magichypercubes.com/Encyclopedia
1960:http://www.magichypercubes.com/Encyclopedia
1718:This positions the number 'k' at position:
522:
351:of the hypercube, and is sometimes denoted
50:Learn how and when to remove these messages
3417:
3403:
3242:this is a n-dimensional version of (pe.):
3170:On the General Properties of Nasik Squares
2908:
2854:Σ ((reflect(k)) ? 2 : 0) ;
2421:expresses and all its r-agonal neighbors.
1478:
861:pandiagonal magic squares parallel to the
782:. In another 1878 paper he showed another
456:, with sequence of magic numbers given by
2896:
2142:
1962:at several places (especially p-section)
617:
288:Learn how and when to remove this message
270:Learn how and when to remove this message
212:Learn how and when to remove this message
147:Learn how and when to remove this message
3183:On the General Properties of Nasik Cubes
3155:Frost, A. H., Invention of Magic Cubes,
2871:
2247:0 ; θ ε {-1,1} > ; p,q ε
2154:
2134:
853:correctly summing lines. They also had 3
187:need to be rewritten in AMS-LaTeX markup
3316:Magic Cubes and Hypercubes - References
2889: : monagonal permutation (axis ε )
2121: : monagonal permutation (axis ε )
1609:{\displaystyle \langle {}_{k}i\rangle }
794:. He referred to all of these cubes as
452:= 4, a magic hypercube may be called a
3638:
3256:Alan Adler magic square multiplication
3254:this is a hyperbeam version of (pe.):
3244:Alan Adler magic square multiplication
2097:
1616:). The method starts at the position P
670:Or, to put it more concisely, all pan-
533:If, in addition, the numbers on every
488:-dimensional magic hypercube of order
3398:
3026:
2388:} might be put in notation as :
1581:) to more general movements (vectors
473:of the magic hypercube is called its
3352:More on this alternative definition.
2526: : the amount of numbers along
1937:'s are combined into the hypercube:
610:numbers passing through each of the
223:
158:
85:adding citations to reliable sources
56:
15:
3322:An algorithm for making magic cubes
2596:
790:lines sum correctly i.e. Hendricks
742:pandiagonal magic squares of order
13:
3424:
3264:
3194:Heinz, H.D., and Hendricks, J.R.,
2917:
2472:, a series that mimics the series
2447:
1809:
1374:
1356:
1292:
786:magic cube and a cube where all 13
778:, and an order-8 cube we class as
14:
3657:
3303:
3196:Magic Square Lexicon: Illustrated
2815:Only one direction with order = 2
2727:all orders are either even or odd
2582:
2320:} : {pan r-agonal; r = 1..n}
2272:
1969:
1921:of radix m numbers (also called "
738:cells. (This cube also contains 9
31:This article has multiple issues.
3473:
3198:, 2000, 0-9687985-0-0 pp 119-122
3157:Quarterly Journal of Mathematics
2938:
2557:: positions within the hyperbeam
2464:generalises the two dimensional
2183:
993:: positions within the hypercube
667:the dimension of the hypercube.
576:is now the preferred term for
228:
163:
61:
20:
3387:Mitsutoshi Nakamura: Rectangles
2774:This is with the exception of m
2763:turns out integer is when all m
2565:: vectors through the hyperbeam
2361:some minor qualifications are:
1891:Latin prescription construction
1834:
1470:
72:needs additional citations for
39:or discuss these issues on the
3248:
3236:
3229:Rosser, B. and Walker, R. J.,
3223:
3214:
3201:
3188:
3175:
3162:
3149:
2767:are even. Thus suffices: all m
2721:
2545:
2492:It is customary to denote the
2487:
2198:
1818:
1803:
1784:
1781:
1702:
1657:
1112:: vector through the hypercube
425:
406:
394:
388:
1:
3515:Prime reciprocal magic square
3382:Marián Trenklar Cube-Ref.html
3318:Collected by Marian Trenkler
3142:
2934:-2 (by monagonal permutation)
2458:n-dimensional magic rectangle
2109: : component permutation
185:the mathematical expressions
3207:Planck, C., M.A., M.R.C.S.,
2826:
2496:with the letter 'n' and the
2403:} can simply be written as:
2313:} : {1-agonal n-agonal}
868:
541:; otherwise, it is called a
7:
3332:Articles by Christian Boyer
3113:
2346:gives arguments for using {
2338:Trump/Boyer {diagonal} cube
1400:{\displaystyle \left=\left}
1236:{\displaystyle \left=\left}
674:-agonals sum correctly for
543:semiperfect magic hypercube
368:, then it has magic number
183:. The specific problem is:
10:
3662:
3312:Articles by Aale de Winkel
3185:, QJM, 15, 1878, pp 93-123
2914:permutation of m numbers.
2468:and the three dimensional
2226:i> ; i ε {-1,0,1}
2055:
1879:gives in his 1905 article
757:
526:
3587:
3561:
3529:Higher dimensional shapes
3528:
3520:Most-perfect magic square
3482:
3471:
3432:
3209:The Theory of Paths Nasik
3172:, QJM, 15, 1878, pp 34-49
2438:most-perfect magic square
1883:The theory of Path Nasiks
1467:of the n numbers 0..n-1.
3574:Pandiagonal magic square
3569:Associative magic square
3510:Pandiagonal magic square
2771:are either even or odd.
2755:When any of the orders m
2587:
2222:i + 1) 3 <==> <
720:pandiagonal magic square
523:Perfect magic hypercubes
500:is different from 2 or
1479:KnightJump construction
659:is the magic constant,
606:(3 − 1) lines of
539:perfect magic hypercube
3338:A magic cube generator
3310:The Magic Encyclopedia
3035:The "normal hyperbeam"
2897:Coördinate permutation
2143:Coordinate permutation
1867:
1780:
1709:
1641:
1610:
1575:
1440:
1401:
1237:
1143:There can be multiple
1106:
987:
847:
833:
618:Nasik magic hypercubes
586:possible lines sum to
580:magic hypercube where
509:R programming language
492:exists if and only if
438:
250:by rewriting it in an
2986: ; i = 0..n-1) (
2930:< ; i = 0..m
2909:Monagonal permutation
2538: − 1.
2344:Nasik magic hypercube
2167:is the special case:
2155:Monagonal permutation
2135:Component permutation
1868:
1754:
1710:
1642:
1640:{\displaystyle V_{0}}
1611:
1576:
1441:
1439:{\displaystyle \left}
1407:("axial"-neighbor of
1402:
1238:
1107:
988:
986:{\displaystyle \left}
837:
822:
625:Nasik magic hypercube
439:
3276:Discrete Mathematics
2674:(m..) abbreviates: m
2603:basic multiplication
2434:Kathleen Ollerenshaw
2350:} as synonymous to {
2333:and divide it by m.
2175:n-agonal permutation
2127: : digit change
1976:basic multiplication
1725:
1654:
1624:
1585:
1487:
1411:
1266:
1153:
999:
880:
375:
194:improve this article
179:to meet Knowledge's
81:improve this article
3610:Eight queens puzzle
3076:"Dynamic numbering"
3068:"Dynamic numbering"
2872:Basic manipulations
2331:Faulhaber's formula
2098:Basic manipulations
511:includes a module,
3336:magichypercube.com
3324:by Marian Trenkler
3279:207 (1999), 65-72.
3159:, 7,1866, pp92-102
3130:Perfect magic cube
3120:Magic cube classes
3070:makes use of the
3027:Special hyperbeams
2831:A hyperbeam knows
2060:A hypercube knows
1863:
1705:
1637:
1606:
1571:
1436:
1397:
1233:
1102:
983:
754:cells, and so on.
529:Magic Cube Classes
434:
315:generalization of
252:encyclopedic style
239:is written like a
3633:
3632:
3579:Multimagic square
3490:Alphamagic square
3137:John R. Hendricks
2924:"normal position"
2864:just as with the
2288:= m (m - 1) / 2.
2086:"normal position"
1814:
1808:
1373:
1355:
1349:
1334:
1331:
1291:
1063:
1024:
944:
905:
519:a multiple of 4.
432:
298:
297:
290:
280:
279:
272:
222:
221:
214:
181:quality standards
172:This article may
157:
156:
149:
131:
96:"Magic hypercube"
54:
3653:
3588:Related concepts
3495:Antimagic square
3477:
3419:
3412:
3405:
3396:
3395:
3258:
3252:
3246:
3240:
3234:
3227:
3221:
3218:
3212:
3205:
3199:
3192:
3186:
3179:
3173:
3166:
3160:
3153:
3090:The "constant 1"
3080:magic hypercubes
2945:magic hypercubes
2866:magic hypercubes
2563:
2562:⟨⟩
2482:magic hypercubes
2430:compact complete
1872:
1870:
1869:
1864:
1856:
1855:
1830:
1829:
1812:
1806:
1802:
1801:
1779:
1768:
1750:
1749:
1737:
1736:
1714:
1712:
1711:
1708:{\displaystyle }
1706:
1701:
1700:
1682:
1681:
1669:
1668:
1646:
1644:
1643:
1638:
1636:
1635:
1615:
1613:
1612:
1607:
1599:
1598:
1593:
1580:
1578:
1577:
1572:
1445:
1443:
1442:
1437:
1435:
1431:
1427:
1426:
1421:
1406:
1404:
1403:
1398:
1396:
1392:
1371:
1353:
1347:
1343:
1342:
1337:
1332:
1329:
1325:
1324:
1319:
1308:
1304:
1289:
1282:
1281:
1276:
1253:is referred to.
1245:Of course given
1242:
1240:
1239:
1234:
1232:
1228:
1224:
1223:
1218:
1209:
1208:
1203:
1192:
1188:
1184:
1183:
1178:
1169:
1168:
1163:
1124:As is indicated
1111:
1109:
1108:
1103:
1101:
1097:
1061:
1022:
1015:
1014:
1009:
992:
990:
989:
984:
982:
978:
942:
903:
896:
895:
890:
845:
831:
722:then would be a
714:
712:
711:
708:
705:
654:
652:
651:
648:
645:
605:
603:
602:
599:
596:
514:
469:The side-length
465:
443:
441:
440:
435:
433:
428:
418:
417:
401:
387:
386:
293:
286:
275:
268:
264:
261:
255:
232:
231:
224:
217:
210:
206:
203:
197:
167:
166:
159:
152:
145:
141:
138:
132:
130:
89:
65:
57:
46:
24:
23:
16:
3661:
3660:
3656:
3655:
3654:
3652:
3651:
3650:
3636:
3635:
3634:
3629:
3605:Number Scrabble
3583:
3557:
3553:Magic hyperbeam
3548:Magic hypercube
3524:
3500:Geomagic square
3478:
3469:
3428:
3423:
3306:
3267:
3265:Further reading
3262:
3261:
3253:
3249:
3241:
3237:
3228:
3224:
3219:
3215:
3206:
3202:
3193:
3189:
3180:
3176:
3167:
3163:
3154:
3150:
3145:
3116:
3105:
3104:
3100:
3092:
3085:
3062:
3058:
3054:
3050:
3049:
3045:
3037:
3029:
3021:
3017:
3013:
3009:
3005:
2993:
2989:
2985:
2966:
2962:
2958:
2954:
2941:
2933:
2920:
2918:normal position
2911:
2899:
2874:
2855:
2853:
2849:
2847:
2843:
2829:
2822:
2817:
2807:
2803:
2799:
2793:
2789:
2785:
2777:
2770:
2766:
2762:
2758:
2750:
2746:
2742:
2738:
2729:
2724:
2717:
2716:
2710:
2709:
2703:
2702:
2696:
2695:
2689:
2685:
2681:
2677:
2670:
2669:
2665:
2659:
2658:
2652:
2651:
2645:
2641:
2639:
2633:
2632:
2628:
2622:
2621:
2615:
2614:
2599:
2590:
2585:
2579:
2578:
2574:
2561:
2548:
2523:
2490:
2466:magic rectangle
2462:magic hyperbeam
2454:magic hyperbeam
2450:
2448:Magic hyperbeam
2417:
2413:
2396:
2392:
2372:
2289:
2287:
2280:
2275:
2270:
2268:
2264:
2260:
2256:
2248:
2246:
2242:
2238:
2227:
2225:
2221:
2217:
2213:
2201:
2186:
2181:
2171:
2157:
2145:
2137:
2100:
2092:
2078:
2076:
2072:
2065:
2058:
2050:
2049:
2042:
2041:
2040:
2036:
2030:
2029:
2023:
2022:
2016:
2012:
2011:
2005:
2004:
2000:
1994:
1993:
1987:
1986:
1972:
1953:
1951:
1947:
1943:
1936:
1928:
1925:"). On these LP
1919:
1917:
1913:
1909:
1905:
1901:
1893:
1851:
1847:
1825:
1821:
1797:
1793:
1769:
1758:
1745:
1741:
1732:
1728:
1726:
1723:
1722:
1690:
1686:
1677:
1673:
1664:
1660:
1655:
1652:
1651:
1631:
1627:
1625:
1622:
1621:
1619:
1594:
1592:
1591:
1586:
1583:
1582:
1488:
1485:
1484:
1481:
1473:
1422:
1420:
1419:
1418:
1414:
1412:
1409:
1408:
1338:
1336:
1335:
1320:
1318:
1317:
1316:
1312:
1277:
1275:
1274:
1273:
1269:
1267:
1264:
1263:
1249:also one value
1219:
1217:
1216:
1204:
1202:
1201:
1200:
1196:
1179:
1177:
1176:
1164:
1162:
1161:
1160:
1156:
1154:
1151:
1150:
1121:
1010:
1008:
1007:
1006:
1002:
1000:
997:
996:
891:
889:
888:
887:
883:
881:
878:
877:
871:
846:
843:
832:
829:
804:who hails from
770:and often call
760:
709:
706:
703:
702:
700:
649:
646:
636:
635:
633:
620:
600:
597:
594:
593:
591:
531:
525:
512:
479:J. R. Hendricks
457:
454:magic tesseract
413:
409:
402:
400:
382:
378:
376:
373:
372:
359:
345:space diagonals
306:magic hypercube
294:
283:
282:
281:
276:
265:
259:
256:
248:help improve it
245:
233:
229:
218:
207:
201:
198:
191:
168:
164:
153:
142:
136:
133:
90:
88:
78:
66:
25:
21:
12:
11:
5:
3659:
3649:
3648:
3631:
3630:
3628:
3627:
3622:
3617:
3615:Magic constant
3612:
3607:
3602:
3597:
3591:
3589:
3585:
3584:
3582:
3581:
3576:
3571:
3565:
3563:
3562:Classification
3559:
3558:
3556:
3555:
3550:
3545:
3544:
3543:
3532:
3530:
3526:
3525:
3523:
3522:
3517:
3512:
3507:
3502:
3497:
3492:
3486:
3484:
3483:Related shapes
3480:
3479:
3472:
3470:
3468:
3467:
3465:Magic triangle
3462:
3457:
3452:
3450:Magic hexagram
3447:
3442:
3436:
3434:
3430:
3429:
3426:Magic polygons
3422:
3421:
3414:
3407:
3399:
3393:
3392:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3333:
3330:multimagie.com
3327:
3326:
3325:
3313:
3305:
3304:External links
3302:
3301:
3300:
3297:
3294:
3291:
3280:
3271:
3266:
3263:
3260:
3259:
3247:
3235:
3222:
3213:
3200:
3187:
3174:
3168:Frost, A. H.,
3161:
3147:
3146:
3144:
3141:
3140:
3139:
3134:
3133:
3132:
3122:
3115:
3112:
3108:
3107:
3102:
3098:
3096:
3091:
3088:
3083:
3064:
3063:
3060:
3056:
3052:
3047:
3043:
3041:
3036:
3033:
3028:
3025:
3024:
3023:
3019:
3015:
3011:
3007:
3003:
2996:
2995:
2991:
2987:
2983:
2969:
2968:
2964:
2960:
2956:
2952:
2940:
2937:
2936:
2935:
2931:
2919:
2916:
2910:
2907:
2898:
2895:
2891:
2890:
2884:
2873:
2870:
2851:
2845:
2841:
2839:
2837:
2828:
2825:
2820:
2816:
2813:
2809:
2808:
2805:
2801:
2797:
2794:
2791:
2787:
2783:
2775:
2768:
2764:
2760:
2756:
2753:
2752:
2748:
2744:
2740:
2736:
2728:
2725:
2723:
2720:
2714:
2712:
2707:
2705:
2700:
2698:
2693:
2691:
2690:abbreviates: m
2687:
2683:
2679:
2675:
2672:
2671:
2667:
2663:
2661:
2656:
2654:
2649:
2647:
2643:
2637:
2635:
2630:
2626:
2624:
2619:
2617:
2612:
2610:
2598:
2597:Multiplication
2595:
2589:
2586:
2584:
2581:
2576:
2572:
2570:
2567:
2566:
2558:
2547:
2544:
2540:
2539:
2519:
2512:
2489:
2486:
2449:
2446:
2428:for squares: {
2423:
2422:
2419:
2415:
2411:
2394:
2390:
2382:
2381:
2374:
2370:
2322:
2321:
2314:
2307:
2300:
2285:
2283:
2278:
2274:
2273:Qualifications
2271:
2266:
2262:
2258:
2254:
2252:
2244:
2240:
2236:
2234:
2223:
2219:
2215:
2211:
2209:
2200:
2197:
2185:
2182:
2179:
2169:
2163:Noted be that
2156:
2153:
2144:
2141:
2136:
2133:
2129:
2128:
2122:
2116:
2110:
2099:
2096:
2090:
2074:
2070:
2068:
2057:
2054:
2047:
2045:
2038:
2034:
2032:
2027:
2025:
2020:
2018:
2014:
2009:
2007:
2002:
1998:
1996:
1991:
1989:
1984:
1982:
1980:
1971:
1970:Multiplication
1968:
1949:
1945:
1941:
1939:
1934:
1931:digit changing
1926:
1915:
1911:
1907:
1903:
1899:
1897:
1892:
1889:
1874:
1873:
1862:
1859:
1854:
1850:
1846:
1843:
1840:
1837:
1833:
1828:
1824:
1820:
1817:
1811:
1805:
1800:
1796:
1792:
1789:
1786:
1783:
1778:
1775:
1772:
1767:
1764:
1761:
1757:
1753:
1748:
1744:
1740:
1735:
1731:
1716:
1715:
1704:
1699:
1696:
1693:
1689:
1685:
1680:
1676:
1672:
1667:
1663:
1659:
1634:
1630:
1617:
1605:
1602:
1597:
1590:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1480:
1477:
1472:
1469:
1463:" specifies a
1434:
1430:
1425:
1417:
1395:
1391:
1388:
1385:
1382:
1379:
1376:
1370:
1367:
1364:
1361:
1358:
1352:
1346:
1341:
1328:
1323:
1315:
1311:
1307:
1303:
1300:
1297:
1294:
1288:
1285:
1280:
1272:
1231:
1227:
1222:
1215:
1212:
1207:
1199:
1195:
1191:
1187:
1182:
1175:
1172:
1167:
1159:
1117:
1114:
1113:
1100:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1021:
1018:
1013:
1005:
994:
981:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
908:
902:
899:
894:
886:
870:
867:
863:space-diagonal
841:
827:
759:
756:
663:the order and
619:
616:
524:
521:
513:library(magic)
446:
445:
431:
427:
424:
421:
416:
412:
408:
405:
399:
396:
393:
390:
385:
381:
355:
349:magic constant
323:, that is, an
296:
295:
278:
277:
236:
234:
227:
220:
219:
171:
169:
162:
155:
154:
69:
67:
60:
55:
29:
28:
26:
19:
9:
6:
4:
3:
2:
3658:
3647:
3646:Magic squares
3644:
3643:
3641:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3592:
3590:
3586:
3580:
3577:
3575:
3572:
3570:
3567:
3566:
3564:
3560:
3554:
3551:
3549:
3546:
3542:
3539:
3538:
3537:
3534:
3533:
3531:
3527:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3487:
3485:
3481:
3476:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3445:Magic hexagon
3443:
3441:
3438:
3437:
3435:
3431:
3427:
3420:
3415:
3413:
3408:
3406:
3401:
3400:
3397:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3337:
3334:
3331:
3328:
3323:
3320:
3319:
3317:
3314:
3311:
3308:
3307:
3298:
3295:
3292:
3289:
3288:0-9687985-0-0
3285:
3281:
3278:
3277:
3272:
3269:
3268:
3257:
3251:
3245:
3239:
3232:
3226:
3217:
3210:
3204:
3197:
3191:
3184:
3181:Frost, A. H.
3178:
3171:
3165:
3158:
3152:
3148:
3138:
3135:
3131:
3128:
3127:
3126:
3123:
3121:
3118:
3117:
3111:
3094:
3093:
3087:
3081:
3077:
3073:
3069:
3039:
3038:
3032:
3001:
3000:
2999:
2981:
2980:
2979:
2976:
2974:
2950:
2949:
2948:
2946:
2939:Qualification
2929:
2928:
2927:
2925:
2915:
2906:
2904:
2894:
2888:
2885:
2882:
2879:
2878:
2877:
2869:
2867:
2863:
2858:
2836:
2834:
2824:
2812:
2795:
2781:
2780:
2779:
2772:
2734:
2733:
2732:
2719:
2608:
2607:
2606:
2605:is given by:
2604:
2594:
2580:
2564:
2559:
2556:
2553:
2552:
2551:
2543:
2537:
2533:
2530:th monagonal
2529:
2525:
2522:
2518:
2513:
2510:
2508:
2503:
2502:
2501:
2499:
2495:
2485:
2483:
2479:
2475:
2471:
2467:
2463:
2459:
2455:
2445:
2441:
2439:
2435:
2431:
2426:
2420:
2410:
2409:
2408:
2406:
2402:
2398:
2387:
2379:
2375:
2368:
2364:
2363:
2362:
2359:
2357:
2353:
2349:
2345:
2341:
2339:
2334:
2332:
2326:
2319:
2315:
2312:
2308:
2305:
2301:
2298:
2294:
2293:
2292:
2282:
2277:A hypercube H
2251:
2233:
2230:
2208:
2206:
2196:
2194:
2190:
2184:Digitchanging
2178:
2176:
2168:
2166:
2161:
2152:
2150:
2140:
2132:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2104:
2103:
2095:
2089:
2087:
2082:
2067:
2063:
2053:
1979:
1978:is given by:
1977:
1967:
1963:
1961:
1957:
1956:J.R.Hendricks
1938:
1932:
1924:
1896:
1888:
1886:
1884:
1878:
1860:
1857:
1852:
1848:
1844:
1841:
1838:
1835:
1831:
1826:
1822:
1815:
1798:
1794:
1787:
1776:
1773:
1770:
1765:
1762:
1759:
1755:
1751:
1746:
1742:
1738:
1733:
1729:
1721:
1720:
1719:
1697:
1694:
1691:
1687:
1683:
1678:
1674:
1670:
1665:
1661:
1650:
1649:
1648:
1632:
1628:
1600:
1595:
1565:
1562:
1559:
1556:
1553:
1547:
1541:
1538:
1535:
1532:
1526:
1520:
1517:
1514:
1511:
1505:
1499:
1496:
1493:
1476:
1468:
1466:
1462:
1457:
1454:
1450:
1447:
1432:
1428:
1423:
1415:
1393:
1389:
1386:
1383:
1380:
1377:
1368:
1365:
1362:
1359:
1350:
1344:
1339:
1326:
1321:
1313:
1309:
1305:
1301:
1298:
1295:
1286:
1283:
1278:
1270:
1261:
1259:
1254:
1252:
1248:
1243:
1229:
1225:
1220:
1213:
1210:
1205:
1197:
1193:
1189:
1185:
1180:
1173:
1170:
1165:
1157:
1148:
1146:
1141:
1139:
1135:
1131:
1127:
1122:
1120:
1098:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1067:
1064:
1058:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1028:
1025:
1019:
1016:
1011:
1003:
995:
979:
972:
969:
966:
963:
960:
957:
954:
948:
945:
939:
933:
930:
927:
924:
921:
918:
915:
909:
906:
900:
897:
892:
884:
876:
875:
874:
866:
864:
860:
856:
852:
840:
836:
826:
821:
819:
815:
811:
807:
803:
799:
798:
793:
789:
785:
781:
777:
773:
769:
765:
755:
753:
749:
745:
741:
737:
733:
729:
725:
721:
716:
698:
694:
689:
687:
686:
681:
677:
673:
668:
666:
662:
658:
643:
639:
631:
627:
626:
615:
613:
609:
589:
585:
584:
579:
575:
574:
569:
565:
561:
557:
556:
550:
548:
545:. The number
544:
540:
536:
535:cross section
530:
520:
518:
510:
505:
503:
499:
495:
491:
487:
482:
480:
476:
472:
467:
464:
460:
455:
451:
429:
422:
419:
414:
410:
403:
397:
391:
383:
379:
371:
370:
369:
367:
363:
358:
354:
350:
346:
342:
338:
334:
330:
326:
322:
318:
317:magic squares
314:
312:
307:
303:
292:
289:
274:
271:
263:
253:
249:
243:
242:
237:This article
235:
226:
225:
216:
213:
205:
195:
190:
188:
182:
178:
177:
170:
161:
160:
151:
148:
140:
129:
126:
122:
119:
115:
112:
108:
105:
101:
98: –
97:
93:
92:Find sources:
86:
82:
76:
75:
70:This article
68:
64:
59:
58:
53:
51:
44:
43:
38:
37:
32:
27:
18:
17:
3625:Magic series
3595:Latin square
3547:
3505:Heterosquare
3455:Magic square
3440:Magic circle
3274:
3250:
3238:
3230:
3225:
3216:
3208:
3203:
3195:
3190:
3182:
3177:
3169:
3164:
3156:
3151:
3109:
3106: : = 1
3065:
3030:
2997:
2977:
2972:
2970:
2942:
2923:
2921:
2912:
2902:
2900:
2892:
2886:
2880:
2875:
2861:
2859:
2856:
2850: ; R =
2832:
2830:
2818:
2810:
2773:
2754:
2730:
2673:
2602:
2600:
2591:
2583:Construction
2568:
2560:
2554:
2549:
2541:
2535:
2531:
2527:
2520:
2516:
2514:
2506:
2504:
2491:
2474:magic square
2469:
2465:
2461:
2457:
2453:
2451:
2442:
2429:
2427:
2424:
2404:
2400:
2389:
2385:
2383:
2377:
2366:
2360:
2355:
2351:
2347:
2342:
2335:
2327:
2323:
2317:
2310:
2304:pan r-agonal
2303:
2296:
2290:
2276:
2249:
2231:
2228:
2204:
2202:
2192:
2189:
2187:
2174:
2172:
2164:
2162:
2158:
2148:
2146:
2138:
2130:
2124:
2118:
2112:
2106:
2101:
2093:
2085:
2083:
2079:
2061:
2059:
2043:
1975:
1973:
1964:
1955:
1954:
1930:
1922:
1920:
1894:
1882:
1876:
1875:
1717:
1482:
1474:
1471:Construction
1461:perm(0..n-1)
1460:
1458:
1455:
1451:
1448:
1262:
1257:
1255:
1250:
1246:
1244:
1149:
1144:
1142:
1137:
1133:
1129:
1125:
1123:
1118:
1115:
872:
858:
854:
850:
848:
838:
834:
823:
812:District in
802:D R Kaprekar
796:
795:
791:
787:
783:
780:pantriagonal
779:
775:
771:
767:
763:
761:
751:
747:
743:
739:
735:
731:
727:
723:
717:
696:
692:
690:
684:
683:
679:
675:
671:
669:
664:
660:
656:
641:
637:
629:
624:
623:
621:
611:
607:
587:
582:
581:
577:
572:
571:
567:
563:
559:
554:
553:
551:
546:
542:
538:
532:
516:
506:
501:
497:
496:> 1 and
493:
489:
485:
483:
474:
470:
468:
453:
449:
447:
365:
361:
356:
352:
336:
332:
328:
324:
313:-dimensional
310:
305:
299:
284:
266:
260:October 2017
257:
238:
208:
199:
192:Please help
184:
173:
143:
137:October 2010
134:
124:
117:
110:
103:
91:
79:Please help
74:verification
71:
47:
40:
34:
33:Please help
30:
3620:Magic graph
3600:Word square
3072:isomorphism
3051: : =
2722:Curiosities
2509:) Dimension
2488:Conventions
2214:where: p =
2205:pathfinders
2199:Pathfinders
2180:_ = _(2-1)
1918:) % m
1465:permutation
1260:= 1 as in:
814:Maharashtra
784:pandiagonal
776:pandiagonal
768:pandiagonal
321:magic cubes
302:mathematics
196:if you can.
3536:Magic cube
3460:Magic star
3391:Peace Cube
3143:References
3125:Magic cube
2534:= 0, ...,
2478:magic cube
2470:magic beam
2405:+ = m - 1
2165:reflection
1459:Further: "
527:See also:
202:March 2014
107:newspapers
36:improve it
3082:of order
2982:S = lcm(m
2903:transpose
2546:Notations
2494:dimension
2149:transpose
1877:C. Planck
1858:−
1845:…
1810:%
1791:∖
1774:−
1756:∑
1695:−
1684:…
1604:⟩
1589:⟨
1569:⟩
1563:−
1554:−
1551:⟨
1545:⟩
1533:−
1530:⟨
1524:⟩
1518:−
1509:⟨
1503:⟩
1491:⟨
1387:−
1375:#
1357:#
1293:#
1089:−
1080:⋯
1068:∈
1050:−
1041:⋯
1029:∈
970:−
961:⋯
949:∈
931:−
922:⋯
910:∈
869:Notations
691:The term
339:array of
42:talk page
3640:Category
3114:See also
3022:- 1) / 2
2994:- 1) / 2
2967:- 1) / 2
2751:- 1) / 2
2623: :
2401:complete
2393:Σ = 2 S
2378:complete
2297:r-agonal
2170:~R = _R
1995: :
1099:⟩
1004:⟨
865:planes.
842:—
828:—
341:integers
335:× ... ×
174:require
3541:classes
2827:Aspects
2682:. (m..)
2524:) Order
2436:called
2407:where:
2386:compact
2367:compact
2356:perfect
2352:perfect
2318:perfect
2269:0 >
2056:Aspects
806:Deolali
792:perfect
772:perfect
758:History
713:
701:
685:perfect
653:
634:
614:cells.
604:
592:
568:perfect
564:perfect
463:A021003
461::
308:is the
246:Please
176:cleanup
121:scholar
3286:
2555:; i= ]
2498:orders
2073:; R =
1923:digits
1813:
1807:
1372:
1354:
1348:
1333:
1330:
1290:
1062:
1023:
943:
904:
825:paper.
678:= 1...
655:where
123:
116:
109:
102:
94:
3433:Types
3101:,..,m
3086:Π m.
3046:,..,m
2973:magic
2823:= 2.
2704:,..,m
2686:(m..)
2678:,..,m
2666:(m..)
2662:(m..)
2655:(m..)
2648:(m..)
2636:(m..)
2629:(m..)
2625:(m..)
2618:(m..)
2611:(m..)
2588:Basic
2575:,..,m
2348:nasik
2311:magic
2253:<
2235:<
2193:perm(
818:India
810:Nasik
797:nasik
764:Nasik
748:nasik
746:.) A
732:nasik
724:nasik
704:3 − 1
693:nasik
573:nasik
475:order
128:JSTOR
114:books
3284:ISBN
2926:by:
2862:n! 2
2373:/ m}
2088:by:
2062:n! 2
1948:Σ LP
1929:'s "
1914:+ LP
1906:Σ LP
1902:= (
507:The
459:OEIS
448:For
319:and
304:, a
100:news
3103:n-1
3084:k=0
3078:of
3059:i m
3053:k=0
3048:n-1
3016:j=0
3008:j=0
3004:max
2988:j=0
2961:j=0
2955:= m
2852:k=0
2846:n-1
2844:..m
2806:m,1
2804:* N
2802:1,m
2800:= N
2792:1,m
2790:* N
2788:m,1
2786:= N
2745:j=0
2739:= m
2713:n-1
2706:n-1
2680:n-1
2640:k=0
2634:= ]
2616:* B
2577:n-1
2412:(k)
2399:. {
2397:/ m
2391:(k)
2265:-1
2218:Σ (
2216:k=0
2191:in
2075:k=0
1988:* H
1946:k=0
1916:k,n
1908:k,l
1904:l=0
808:in
583:all
578:any
560:all
300:In
83:by
3642::
3055:Σ
3018:Πm
3010:Πm
3006:=
2990:Πm
2975:}
2963:Πm
2887:_2
2840:(m
2747:Πm
2718:.
2653:+
2644:k1
2642:Πm
2476:,
2452:A
2418:1.
2261:1
2257:1
2243:θ
2239:1
2210:Pf
2119:_2
2024:+
2006:=
1952:m
1944:=
1898:LP
1861:1.
1446:)
816:,
718:A
715:.
699:=
644:+1
632:=
622:A
570:,
481:.
466:.
331:×
327:×
45:.
3418:e
3411:t
3404:v
3290:.
3099:0
3097:m
3095:1
3061:k
3057:k
3044:0
3042:m
3040:N
3020:j
3014:(
3012:j
3002:S
2992:j
2984:i
2965:j
2959:(
2957:k
2953:k
2951:S
2932:k
2881:^
2848:)
2842:0
2838:B
2833:2
2821:k
2798:m
2796:N
2784:m
2782:N
2776:k
2769:k
2765:k
2761:k
2757:k
2749:j
2743:(
2741:k
2737:k
2735:S
2715:2
2711:m
2708:1
2701:2
2699:0
2697:m
2694:1
2692:0
2688:2
2684:1
2676:0
2668:2
2664:1
2660:]
2657:2
2650:2
2646:]
2638:1
2631:2
2627:1
2620:2
2613:1
2609:B
2573:0
2571:m
2536:n
2532:k
2528:k
2521:k
2517:m
2515:(
2507:n
2505:(
2456:(
2416:k
2395:m
2384:{
2376:{
2371:m
2365:{
2316:{
2309:{
2302:{
2295:{
2286:m
2284:S
2279:m
2267:s
2263:l
2259:k
2255:j
2245:l
2241:k
2237:j
2224:k
2220:k
2212:p
2125:=
2113:^
2107:#
2071:m
2069:H
2048:2
2046:m
2039:2
2037:m
2035:1
2033:m
2031:]
2028:2
2026:m
2021:2
2019:m
2017:]
2015:1
2013:m
2010:1
2008:m
2003:2
2001:m
1999:1
1997:m
1992:2
1990:m
1985:1
1983:m
1981:H
1950:k
1942:m
1940:H
1935:k
1927:k
1912:l
1910:x
1900:k
1885:"
1881:"
1853:n
1849:m
1842:0
1839:=
1836:k
1832:;
1827:l
1823:V
1819:)
1816:m
1804:)
1799:l
1795:m
1788:k
1785:(
1782:(
1777:1
1771:n
1766:0
1763:=
1760:l
1752:+
1747:0
1743:P
1739:=
1734:k
1730:P
1703:]
1698:1
1692:n
1688:V
1679:0
1675:V
1671:,
1666:0
1662:P
1658:[
1633:0
1629:V
1618:0
1601:i
1596:k
1566:2
1560:,
1557:1
1548:,
1542:2
1539:,
1536:1
1527:,
1521:2
1515:,
1512:1
1506:,
1500:2
1497:,
1494:1
1433:]
1429:0
1424:k
1416:[
1394:]
1390:1
1384:n
1381:=
1378:j
1369:;
1366:1
1363:=
1360:k
1351:;
1345:0
1340:j
1327:1
1322:k
1314:[
1310:=
1306:]
1302:1
1299:=
1296:k
1287:;
1284:1
1279:k
1271:[
1258:k
1251:i
1247:k
1230:]
1226:i
1221:1
1214:,
1211:j
1206:k
1198:[
1194:=
1190:]
1186:j
1181:k
1174:,
1171:i
1166:1
1158:[
1145:k
1138:m
1134:i
1130:i
1126:k
1119:m
1095:}
1092:1
1086:m
1083:,
1077:,
1074:0
1071:{
1065:i
1059:;
1056:}
1053:1
1047:n
1044:,
1038:,
1035:0
1032:{
1026:k
1020:;
1017:i
1012:k
980:]
976:}
973:1
967:m
964:,
958:,
955:0
952:{
946:i
940:;
937:}
934:1
928:n
925:,
919:,
916:0
913:{
907:k
901:;
898:i
893:k
885:[
859:m
855:m
851:m
788:m
752:m
744:m
740:m
736:m
728:m
710:2
707:/
697:P
680:n
676:r
672:r
665:n
661:m
657:S
650:2
647:/
642:m
640:(
638:m
630:S
612:m
608:m
601:2
598:/
595:1
588:S
547:n
517:n
502:p
498:n
494:p
490:n
486:p
471:n
450:k
444:.
430:2
426:)
423:1
420:+
415:k
411:n
407:(
404:n
398:=
395:)
392:n
389:(
384:k
380:M
366:n
362:n
360:(
357:k
353:M
337:n
333:n
329:n
325:n
311:k
291:)
285:(
273:)
267:(
262:)
258:(
254:.
215:)
209:(
204:)
200:(
189:.
150:)
144:(
139:)
135:(
125:·
118:·
111:·
104:·
77:.
52:)
48:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
