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Dudeney Patterns
{note: investigative article}
(by Aale de Winkel)



H.E. Dudeney (Amusements in Mathematics, 1917, p120) recognized 12 different complementairy pair patterns within the order 4 magic squares. This article represents them in an analitic manner, which provides means to generalize. With thanks to <Harvey Heinz> who finds his dudeney pattern pictures of his transform page reused here. Though I transposed the pictures of the groups XI and XII for the explained reason.
This article provides the basis for the "patterns article", I'll intend to write shortly. Currently the remarks section mentions some unclear things which will need some futher investigation.

The Magic Encyclopedia
Dudeney Patterns
(note: transposed versions are also found in order 4 squares)
pattern horizontal vertical

I
2,3,0,1 2,3,0,1

II
1,0,3,2 1,0,3,2

III
3,2,1,0 3,2,1,0

IV
0,1,2,3 1,0,3,2

V
0,1,2,3 2,3,0,1

VI
0,1,2,3 3,2,1,0

VII
0,1,2,3 {(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}

VIII
0,1,2,3 {(2,3,0,1),(3,2,1,0),(3,2,1,0),(2,3,1,0)}

IX
0,1,2,3 {(1,0,3,2),(3,2,1,0),(3,2,1,0),(1,0,3,2)}

X
0,1,2,3 {(3,2,1,0),(2,3,0,1),(2,3,0,1),(3,2,1,0)}

XI
1,0,3,2 0,3,2,1

XII
2,3,0,1 0,1,3,2
Theoretical basis
The use of "self inverse permutations" to desribe complementairy pairs within a square gives us some
analitic handles to describe the pair. The combination of a horizontal and vertical permutation describe
a pair of cells where complementairy pairs reside provided that the horizontal (H) and vertical (V)
obey the simple rule H[i] <> V[i] (otherwise the combination would map a cell onto itself, and not
describe a cell-pair), From the above listed Dudeney patterns, show that most patterns can be described
by only a single permutation in either direction, the patterns VII / X every column need a seperate
permutation to describe the involved pairs, although noticed the central vertical symmetry which might
be utilized (Left for some future date)
self inverse permutation
factor: 10
Permutation which is it's own inverse ie: P[P[i]] = i
There are 3 possibilities for pairing off the first element for a swap, with
the other pair either in normal or reversed order
when the first element is not involved there are (32) = 3
posibilities for swapping pairs aside from the neutral.
(see p-tuple patterns article for generalisation of this argument)
(0,1,2,3)
(0,1,3,2) , (0,2,1,3) , (0,3,2,1)
(1,0,2,3) , (2,1,0,3) , (3,1,2,0)
(1,0,3,2) , (2,3,0,1) , (3,2,1,0)
remarks
Notice that the dudeney patterns all use merely the symmetric permutations in the above shown horizontal
direction, ie the permutations (0,1,2,3) , (0,2,1,3) , (3,1,2,0) , (1,0,3,2) , (3,2,1,0) and (2,3,0,1)
in fact only patterns XI uses tha asymmetric (0,3,2,1) and pattern XII the asymmetric (0,1,3,2)
for some (yet unclear) reason the permutations (0,2,1,3), (1,0,2,3), (2,1,0,3) and (3,1,2,0) seem
to be missing from the dudeney patterns, so the used permutations are:
(0,1,2,3) , (0,1,3,2) , (0,3,2,1) , (1,0,3,2) , (3,2,1,0) and (2,3,0,1)
note that (1,0,2,3) and (2,1,0,3) are mirrors of (0,1,3,2) and (0,3,2,1) respectively, and
result in just horizontal mirriors of dudeney patterns XI and XII when used.
while (0,2,1,3) is outer nutral and (3,1,2,0) is inner nutral, suggesting it is neccessairy
to do bothe (ie (3,2,1,0)) or swap at least 1 inner to an outer position

Note the above remarks need further investigation, it might lead to some theory behind complementairy pair patterns, simular means can provide some insight in more general p-tuple patterns, concideration I'll leave for a more general discussion.

Recent interest in the study of complementairy pairs in the higher dimensions lead me to postulate the following more systematized viewpoint, in honour of Dudeney the DP stand for Dudeney Pattern. For this discussion the patterns VII, VIII, IX and X are left out for simplicity but can simularly be added (might do so in future upload).

The Magic Encyclopedia
Dudeney Patterns
(note: generalisation of order 4 Dudeney Patterns in higer dimension)
description remark
DP{[0,1,2,3],[0,1,2,3]} Nutral mapping of cells onto themselfs
Patterns involving a single 1-agonal
(note: IV (transposed) is isued as a sample (the other patterns can simularly be transposed)
DP{[1,0,3,2],[0,1,2,3]} pairing of {{0,y],[0,y]} and {{2,y],[3,y]}
Dudeney Pattern IV (transposed)
DP{[0,1,2,3],[1,0,3,2]} pairing of {{x,0],[x,1]} and {{x,2],[x,3]}
Dudeney Pattern IV
DP{[0,1,2,3],[2,3,0,1]} pairing of {{x,0],[x,2]} and {{x,1],[x,3]}
Dudeney Pattern V
DP{[0,1,2,3],[3,2,1,0]} pairing of {{x,0],[x,3]} and {{x,1],[x,2]}
Dudeney Pattern V1
Patterns involving 2 1-agonals
DP{[1,0,3,2],[1,0,3,2]} pairing of {{0,0],[1,1]} and {{2,2],[3,2]} and mixed
Dudeney Pattern II
DP{[2,3,0,1],[2,3,0,1]} Dudeney Pattern I
DP{[3,2,0,1],[3,2,0,1]} Dudeney Pattern III
DP{[1,0,3,2],[0,3,2,1]} Dudeney Pattern X1
DP{[2,3,0,1],[0,1,3,2]} Dudeney Pattern X1I
adding dimension is simply done by inserting a selfinverse permutation in a "slot"
opened by that dimension. the added dimension is as usual opening a "slot" on the right
movein the permutation to another "slot" gives merely a transpositional variant
DP{[1,0,3,2],[1,0,3,2],[0,1,2,3]} Dudeney Pattern II in every x-y plane
DP{[1,0,3,2],[1,0,3,2],[1,0,3,2]} Dudeney Pattern II in every x-y y-z and x-z planes
DP{[1,0,3,2],[1,0,3,2],[2,3,0,1]} Dudeney Pattern II in every x-y plane
Dudeney Pattern I in every x-z and y-z plane
DP{[1,0,3,2],[1,0,3,2],[3,2,1,0]} Dudeney Pattern II in every x-y plane
Dudeney Pattern III in every x-z and y-z plane
With the above notation regular camplementairy pair patterns can be handled
however aside from seeing the square dudeny pattern when the (hyper)cube is viewed
perpendicular to an orthogonal plane the above already show samples where in one
direction there is no pairing at all (pe PD{[0,1,2,3],[0,1,2,3],[1,0,3,2]} which
shows the dudeney IV pattern entirily in the z direction in either the x-z and y-z
planes, while the x-y planes show no paring at all. Reflecting upon this any combination
of the 10 possible self-inverse permutations can be used in either x or y direction
giving 102 posibilities combined with either [1,0,3,2], [2,3,0,1] or [3,2,1,0]
in the z-direction. More complex combination in the z-direction are possible given
a permutation pair on the x-y plane.
Functional pair parametrisation
Dispite the complex pair patterns which can be handled with the above method, already in
the order 4 cube quite regular patterns can be found which defy this notation. When pe the
position [i,j,k] paired of with position [3-k,3-j,3-i] defies the notation. This twisted
Dudeney VI pattern might be denotable by DP{[3,2,1,0][2,1,0]} as well
as DP{[3-z,3-y,3-x]} (probably however the latter is less general)
NOTE: all this is preliminairy pending future investigation, (feel free to contribute)

<Walter Trump>'s research alerted us once more that squares are isomorphic to one another, epescially those within Dudeney subgroups. Hence every pair of dudeney pictures denotes a isomorphism. Halve of these cells remain constant while the complementairy halve move around in the square. We thus have (122) complementairy element isomorphisms denotable by two dudeney subgroups. A row or column swap can of course be denoted by a permutation (pe [0,2,1,3] swaps row/column 1 with row/column) with the understanding that when say two rows are swapped by the first permutation the second permutation acts only on those moved by the first, a permutation pair can denote the complete isomorphism (thus IM{[0,2,1,3],[0,2,1,3]} is abbreviating the more complex IM{[0,2,1,3],{[0,1,2,3], [0,2,1,3],[0,2,1,0],[0,1,2,3]}}. Below a fraction of the above mentioned 66 isomorphisms is shown. The shown description of the isomorphsim is but a possibility, though quite probably other possibilities are releted to the one shown. The isomorphism between group II and IV exist, however seems to fall outside the posibilities of the described formalism. Also groups XI and XII suffer the same fate, as groups VII till X one need to use the more elaborate version. The table below shows that the 12 dudeney patterns thus split into four groups (I,II,III), (IV,V,VI), (VII,VIII,IX,X) and (XI,XII)
The square of positions show the new positions of the "Normal square" N4 after application of the isomorphism. This manner of describing an isomorphism is more general and allows also to describe the isomorphisms between the four groups.

Dudeney Pattern Isomorphisms
patterns isomorphism
group II ==> group I

DP{[1,0,3,2],[1,0,3,2]} ==> DP{[2,3,0,1],[2,3,0,1]}
IM{[0,2,1,3],[2,3,0,1]}

[0,0] [2,3] [1,3] [3,0]
[0,1] [2,2] [1,1] [3,1]
[0,2] [2,1] [1,1] [3,2]
[0,3] [2,0] [1,0] [3,3]
group II ==> group III

DP{[1,0,3,2],[1,0,3,2]} ==> DP{[3,2,1,0],[3,2,1,0]}
IM{[0,3,2,1],[2,3,0,1]}

[0,0] [3,2] [2,0] [1,2]
[0,1] [3,3] [2,1] [1,3]
[0,2] [3,0] [2,2] [1,0]
[0,3] [3,1] [2,3] [1,1]
group IV ==> group V

DP{[0,1,2,3],[1,0,3,2]} ==> DP{[0,1,2,3],[2,3,0,1]}
IM{[0,1,2,3],[0,2,1,3]}

[0,0] [1,0] [2,0] [3,0]
[0,2] [1,2] [2,2] [3,2]
[0,1] [1,1] [2,1] [3,1]
[0,3] [1,3] [2,3] [3,3]
group IV ==> group VI

DP{[0,1,2,3],[1,0,3,2]} ==> DP{[0,1,2,3],[3,2,1,0]}
IM{[0,1,2,3],[0,3,2,1]}

[0,0] [1,0] [2,0] [3,0]
[0,3] [1,3] [2,3] [3,3]
[0,2] [1,2] [2,2] [3,2]
[0,1] [1,1] [2,1] [3,1]
group VII ==> group VIII

DP{[0,1,2,3],{(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}} ==>
DP{[0,1,2,3],{(2,3,1,0),(3,2,1,0),(3,2,1,0),(2,3,1,0)}}
IM{[0,1,2,3],{[0,1,3,2],[0,3,2,1],[0,3,2,1],[0,1,3,2]}

[0,0] [1,0] [2,0] [3,0]
[0,1] [1,3] [2,3] [3,1]
[0,3] [1,2] [2,2] [3,3]
[0,2] [1,1] [2,1] [3,2]
group VII ==> group IX

DP{[0,1,2,3],{(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}} ==>
DP{[0,1,2,3],{(1,0,3,2),(3,2,1,0),(3,2,1,0),(1,0,3,2)}}
IM{[0,1,2,3],[0,3,2,1]}

[0,0] [1,0] [2,0] [3,0]
[0,3] [1,3] [2,3] [3,3]
[0,2] [1,2] [2,2] [3,2]
[0,1] [1,1] [2,1] [3,1]
group VII ==> group X

DP{[0,1,2,3],{(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}} ==>
DP{[0,1,2,3],{(3,2,1,0),(2,3,0,1),(2,3,0,1),(3,2,1,0)}}
IM{[0,1,2,3],{[0,1,2,3],[0,2,1,3],[0,2,1,3],[0,1,2,3]}

[0,0] [1,0] [2,0] [3,0]
[0,1] [1,2] [2,2] [3,1]
[0,2] [1,1] [2,1] [3,2]
[0,3] [1,3] [2,3] [3,3]
group XI ==> group XII

DP{[1,0,3,2],[0,3,2,1]} ==> DP{[2,3,0,1],[0,1,3,2]}
IM{[0,3,2,1],[0,2,3,1]}

[0,0] [2,0] [1,0] [3,0]
[0,1] [2,3] [1,3] [3,1]
[0,2] [2,1] [1,1] [3,2]
[0,3] [2,2] [1,2] [3,3]
group II ==> group IV

DP{[1,0,3,2],[1,0,3,2]} ==> DP{[0,1,2,3],[1,0,2,3]}
[0,0] [1,0] [2,0] [3,0]
[1,1] [0,1] [3,1] [2,1]
[1,2] [0,2] [3,2] [2,2]
[0,3] [1,3] [2,3] [3,3]
group II ==> group VII

DP{[1,0,3,2],[1,0,3,2]} ==> DP{[0,1,2,3],{(3,2,1,0),(1,0,3,2),(1,0,3,2),(3,2,1,0)}}
[0,0] [1,0] [2,0] [3,0]
[1,3] [0,1] [3,1] [2,3]
[0,2] [1,2] [2,2] [3,2]
[1,1] [0,3] [3,3] [2,1]
group II ==> group XI

DP{[1,0,3,2],[1,0,3,2]} ==> DP{[1,0,3,2],[0,3,2,1]}
[0,0] [1,1] [2,1] [3,0]
[0,1] [1,2] [2,2] [3,1]
[0,2] [1,2] [2,2] [3,2]
[0,3] [1,0] [2,0] [3,3]