The Magic Encyclopedia ™

Group Theory

<Francis Gaspalou> discusses the grouptheoretical aspects of the magic squares in magic-squares

This file list some results based on my own investigation based on <Gil Lamb>'s Generating Squares (see elsewhere in this Encyclopedia)

Dynamic numbering forms a groupoperator with the generating squares as groupgenerators. The order of a group is the amount of elements within the group, this becomes a bit confusing with the regular meaning of the word 'order', so I'll use 'grouporder' where the order of the group is meant.

Most elements of the GS-groups here when dynamically numbered with repeatedly itself will map onto the identity N

(note: I said 'most' since I cut off my routine at p = 1000, and order 12 elements I found which do not seem to display the feature or is well beyond this cutoff element like:

Order 4 {2-pan} squares | ||
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Aside from the 48 {pan}magic squares there are 384 (8*48) {2-pan}magic squares The 3 order 4 Generating Squares {N _{4},_{2}N_{1}*_{2}N_{4},_{2}N_{2}*_{2}N_{2}}forms a dynamic numbering group of 'group-order 6' (ie 6 elements 'GS') which combines any of the {2-pan} squares S with 5 others {GS}S elements {GS}({GS}S) are elements that are already known Further the 'panrelocation' @[2,1] combined with the "column-panflip' _1[0,3,2,1] (ie @[2,1]_1[0,3,2,1]) traverses the set of squares in four steps, and can be combined with the 'diagonal-panflip' _3[0,3,2,1] into a group of 'grouporder 8' Combining the GS-group with the 'panflip-group' combines into a "grouporder 48' group {GS} . {@[2,1]_1[0,3,2,1]} . {_3[0,3,2,1]} | ||

N_{4}(power: 1) 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 |
_{2}N_{1}*_{2}N_{4}(power: 3) 00 01 08 09 02 03 10 11 04 05 12 13 06 07 14 15 |
N_{2}*N_{2}(power: 2) 00 01 04 05 02 03 06 07 08 09 12 13 10 11 14 15 |

{_{2}N_{1}*_{2}N_{4}}(_{2}N_{1}*_{2}N_{4})(power: 3) 00 01 04 05 08 09 12 13 02 03 06 07 10 11 14 15 |
{N_{2}*N_{2}}(_{2}N_{1}*_{2}N_{4})(power: 2) 00 01 08 09 04 05 12 13 02 03 10 11 06 07 14 15 |
{_{2}N_{1}*_{2}N_{4}}(N_{2}*N_{2})(power: 2) 00 01 02 03 08 09 10 11 04 05 06 07 12 13 14 15 |

Further of course (@[2,1]_1[0,3,2,1) ^{4} = @[0,0]_1[0,1,2,3] and(_3[0,3,2,1]) ^{2} = _3[0,1,2,3]The forementioned groupgenerators form a seperation of the 384 {2-pan} squares in 8 seperate groups the frirst elements are squares 16, 18, 21, 27, 30, 32, 34 and 73 as said also the {pan}squares are interconnected by this same group with square 102 as the first element in the database uploaded ordered listing, though being a group any square within the group can be used and the square "Pan" forms the easiest constructable group element of the {pan}magic squares by the "Pan-transform". | ||

Order 8 Dynamic numbering group | ||

The 'grouporder' of the group with 10 generators is yet undertermined. N _{8} (power: 1)_{2}N_{1}*_{4}N_{8} (power: 4)_{4}N_{1}*_{2}N_{8} (power: 5)N _{2}*N_{4} (power: 3)_{4}N_{2}*_{2}N_{4} (power: 2)_{2}N_{1}*N_{2}*_{2}N_{4} (power: 6)_{2}N_{4}*_{4}N_{2} (power: 2)N _{4}*N_{2} (power: 3)_{2}N_{1}*_{2}N_{4}*N_{2} (power: 5)N _{2}*N_{2}*N_{2} (power: 4)
| ||

The 100 elements {GS}GS has 21 known elements since {N _{8}}GS = {GS}N_{8} = GS and{N _{2}*N_{4}}(_{4}N_{2}*_{2}N_{4}) = N_{4}*N_{2}{N _{2}*N_{4}}(_{2}N_{1}*_{2}N_{4}*N_{2}) = _{4}N_{1}*_{2}N_{8}{ _{4}N_{2}*_{2}N_{4}}(_{4}N_{2}*_{2}N_{4}*N_{2}) = N_{8}{ _{2}N_{4}*_{4}N_{2}}(_{4}N_{2}*_{2}N_{4}) = N_{2}*N_{2}*N_{2}{ _{2}N_{4}*_{4}N_{2}}(_{2}N_{1}*N_{2}*_{2}N_{4}) = _{2}N_{1}*_{2}N_{4}*N_{2}{ _{2}N_{4}*_{4}N_{2}}(_{2}N_{4}*_{4}N_{2}) = N_{8}{ _{2}N_{4}*_{4}N_{2}}(_{2}N_{1}*_{2}N_{1}*_{2}N_{4}*N_{2}) = _{2}N_{1}*N_{2}*_{2}N_{4}{ _{2}N_{4}*_{4}N_{2}}(N_{2}*N_{2}*N_{2}) = _{4}N_{2}*_{2}N_{4}{N _{4}*N_{2}}(_{4}N_{2}*_{2}N_{4}) = N_{2}*N_{4}{N _{4}*N_{2}}(_{2}N_{1}*N_{2}*_{2}N_{4}) = _{2}N_{1}*_{4}N_{8}{N _{2}*N_{2}*N_{2}}(_{4}N_{2}*_{2}N_{4}) = _{2}N_{4}*_{4}N_{2}
the rest are all new elements of the group ("power" ranges from 1 till 7) | ||

Application on the square "Pan" of elements of the above mentioned order 8 DN-group reveiled only {pan}magic squares, a strong indication that this order 8 group is invariant for the {panmagic} qualification, it is NOT a {bimagic} invariant. Only a few elements dynamically numbered the tested bimagic square into a bimagic square since this did not continue with reapplication of the same element this doen't count | ||

Order 9 Dynamic numbering group | ||

<Gil Lamb> discovered the dynamically
numbering the order 9 Collison bimagic square he obtained yet another bimagic square, this seems to continue for all order 9 bimagic squares. and not only the {3-pan bimagic 3-panmagic} squares the Collison square is: (square #184) _{_3[1,6,3,0,4,8,5,2,7]}LP({{1,1,0,2},{2,1,1,0}},{{0,1,1,2}{2,0,1,1}} _{_[7,2,3,5,6,1,0,4,8]}^{t})
_{_3[1,6,3,0,4,8,5,2,7]}
| ||

The 'grouporder' of this group is the same as for order 4 thus 6 elements: N _{9}_{3}N_{1}*_{3}N_{9}N _{3}*N_{3}{ _{3}N_{1}*_{3}N_{9}}(_{3}N_{1}*_{3}N_{9}){ _{3}N_{1}*_{3}N_{9}}(N_{3}*N_{3}){N _{3}*N_{3}}(_{3}N_{1}*_{3}N_{9})
| ||

Order 12 Dynamic numbering group | ||

The vast order 12 DN-group with 41 generators shows elements that are not {panmagic} invariants most elements seem to be invariants, a subgroup might be definable that is {panmagic} invariant |

Currently I have no idea why the order 9 DN-group works for the bimagic squares, also the thus found {3-pan bimagic 3-panmagic} squares seems not to be obtainable by the tri-digital equations with which I obtained the database listed squares. As above metioned only a few elements worked on the order 8 {4-pan bimagic panmagic} square I tested. The groups seem to be a {panmgic} invariant, though the order 12 evidence indicates tat one might need to exclude some elements as {{

from Small_Latin_squares_and_quasigroups I learned that there are but 4 reduced latin squares

Order 4 group theory | |||
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RLS_A abcd 0123 badc 1032 cdab 2301 dcba 3210 |
RLS_B abcd 0123 badc 1032 cdba 2310 dcab 3201 |
RLS_C abcd 0123 bcda 1230 cdab 2301 dabc 3012 |
RLS_D abcd 0123 bdac 1302 cadb 2031 dcba 3210 |

which expands to all 576 possibilities by performing 6 row permutations of type _2[0,perm(1,2,3)] and the 24 column permutations _1[perm(0,1,2,3)], further associating a,b,c,d with 0,1,2,3 gives the normally used form of latin squares on this site also shown above examination learns only RLS_A used in order 4 {panmagic) squares | |||

RLS_A_{_2[0,2,1,3]}0123 2301 1032 3210 |
RLS_A_{_2[0,1,3,2]=[0,3,1,2]}0312 3021 2130 1203 |
RLS_A_{_2[0,3,1,2]}0123 3210 1032 2301 |
RLS_A_{_2[0,2,3,1]=[0,3,1,2]}0312 1203 2130 3021 |

Note that the latter 2 are related to the first 2 by the vertical "panflip" _2[0,3,2,1] while of course the first 2 are related by the combined _2[0,2,3,1]=[0,3,1,2] (!?) also see that RLS_A _{_2[0,3,1,2]} = RLS_A_{_2[0,2,3,1]=[0,3,1,2]}^{t} making thefirst a bit more natural then the latter since 4 * RLS_A _{_2[0,3,1,2]} + RLS_A_{_2[0,3,1,2]}^{t} isin it's normalized position immediately. As this LS has no equal elements symmetric to the maindiagonal it is orthogonal to it's transposed and thus all numbers are present in this square (the square thus obtained is #104 in my uploaded list which is but the vertical panflip of the first panmagic square (ie #104 = #102 _{_2[0,3,2,1]}) which serves as entrypoint forthe dynamic numbering-panflip group of grouporder 48 mentioned in the previous section) | |||

Note: the table above is a novel approach, it will be augmented as time permits