The Magic Encyclopedia ™

Group Theory
(by Aale de Winkel)

<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 Nm (as I currently understand nomenclature) this forms an (of course) abelean (single generator) DN-subgroup of grouporder p where: GSp = Nm (below mentioned as: power p)
(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: 3N1*2N6*N2)

Order 4 {2-pan} squares
Aside from the 48 {pan}magic squares there are 384 (8*48) {2-pan}magic squares
The 3 order 4 Generating Squares {N4,2N1*2N4,2N2*2N2}
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]}
N4
(power: 1)

00 01 02 03
04 05 06 07
08 09 10 11
12 13 14 15
2N1*2N4
(power: 3)

00 01 08 09
02 03 10 11
04 05 12 13
06 07 14 15
N2*N2
(power: 2)

00 01 04 05
02 03 06 07
08 09 12 13
10 11 14 15
{2N1*2N4}(2N1*2N4)
(power: 3)

00 01 04 05
08 09 12 13
02 03 06 07
10 11 14 15
{N2*N2}(2N1*2N4)
(power: 2)

00 01 08 09
04 05 12 13
02 03 10 11
06 07 14 15
{2N1*2N4}(N2*N2)
(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.

N8 (power: 1)
2N1*4N8 (power: 4)
4N1*2N8 (power: 5)
N2*N4 (power: 3)
4N2*2N4 (power: 2)
2N1*N2*2N4 (power: 6)
2N4*4N2 (power: 2)
N4*N2 (power: 3)
2N1*2N4*N2 (power: 5)
N2*N2*N2 (power: 4)
The 100 elements {GS}GS has 21 known elements since
{N8}GS = {GS}N8 = GS and

{N2*N4}(4N2*2N4) = N4*N2
{N2*N4}(2N1*2N4*N2) = 4N1*2N8
{4N2*2N4}(4N2*2N4*N2) = N8
{2N4*4N2}(4N2*2N4) = N2*N2*N2
{2N4*4N2}(2N1*N2*2N4) = 2N1*2N4*N2
{2N4*4N2}(2N4*4N2) = N8
{2N4*4N2}(2N1*2N1*2N4*N2) = 2N1*N2*2N4
{2N4*4N2}(N2*N2*N2) = 4N2*2N4
{N4*N2}(4N2*2N4) = N2*N4
{N4*N2}(2N1*N2*2N4) = 2N1*4N8
{N2*N2*N2}(4N2*2N4) = 2N4*4N2

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:

N9
3N1*3N9
N3*N3
{3N1*3N9}(3N1*3N9)
{3N1*3N9}(N3*N3)
{N3*N3}(3N1*3N9)
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 {{2N1*6N12}(N3*N4)} Pan is not a panmagic square among others



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

Order 4 group theory
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 the
first a bit more natural then the latter since 4 * RLS_A_2[0,3,1,2] + RLS_A_2[0,3,1,2]t is
in 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 for
the 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