The Magic Encyclopedia ™

P-Tuple 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 generalizes this idea to p-tuple pattern on any order.
Currently only complementairy pairs are shown, but this will be generalized to p-tuples on future uploads

The Magic Encyclopedia
P-tuple Patterns

Theoretical basis (complementairy pairs)
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 denote
cells within a square. In case both permutation are in natural order they act like normal cell-coordinates
The single horizontal (H) and vertical permutation (V) this article uses intend to be such that no cell is
mapped onto itself (ie: (i,j) <-> (H[i],V[j]) H[i] <> i or V[j] <> j for i,j = 0 .. m-1)
self inverse permutation
factor:
F(1) = 1
F(2) = 1 + (²₂) = 2
F(3) = 1 + (³₂) = 4
F(4) = 3 F(2) + F(3) = 10

F(m) = (m-1) F(m-2) + F(m-1)
Permutation which is it's own inverse ie: P[P[i]] = i
The first permutation element is involved in m-1 swaps, the other
elements for obvious reason obey the order m-2 formula. When the
first element is not involved the m-1 other elements are subject
to the order m-1 argumentation
(0) (0,1,2,3)
(0,1,3,2)
(0,2,1,3)
(0,3,1,2)

(1,0,2,3)
(1,0,3,2)
(2,1,0,3)
(2,3,0,1)
(3,1,2,0)
(3,2,1,0)

(0,1)
(1,0)
(0,1,2)
(0,2,1)
(1,0,2)
(2,1,0)

Diagonal symmetric patterns pattern symmmetric in main diagonal
With permutations such that P[i] <> i, the permutation can be used
in both directions
(Dudeney pattern I, II and III are of this type (see Dudeney article))

1-agonal invariant paterns pattern swapping elements within 1 1-agonal
With permutations such that P[i] <> i, the permutation can be used
in one of the directions while the other direction the natural order
is maintained (ie combined with P[i] == i)
(Dudeney patterns IV / X are of this type (see Dudeney article))

central 1-agonal symmetric patterns patttern symmetric in central 1-agonal
With symmetric permutations (ie: P[i] + P[m-1-i] = m-1), used in
one of the directions the pattern is symmetric in the central
perpendicuar direction
(all Dudeney patterns are of this type (see Dudeney article))

Theoretical basis (p-tuple permutations)
(preliminairy)
p-tuple permutation permutation which whe applied p times results in the identity
permutations like (1,2,3,0) applied 4 time result in the identity:
(1,2,3,0)(1,2,3,0)(1,2,3,0)(1,2,3,0) = (1,2,3,0)(1,2,3,0)(2,3,0,1) =
(1,2,3,0)(3,0,1,2) = (0,1,2,3)
These kind of permutations applied in the same manner in 1-agonal directions
as the self-inverse permutations lays relations between (p=4) cells

Theoretical basis (pattern desccription with p-tuple permutations)
(preliminairy)
patterns use of multipe p-tuple permutations can describe all kinds of patterns
Once expertise is obtained (by programmatic experimentation)
the usefullness of this avenue can be ascertained
(pe {(1,2,3,4,0),(4,3,1,0)} pair in both direction define an order 5 X pattern)

Currently this preliminairy notes, will be augmented if time permits

The Magic Encyclopedia P-tuple Patterns
Theoretical basis (complementairy pairs)
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 denote cells within a square. In case both permutation are in natural order they act like normal cell-coordinates The single horizontal (H) and vertical permutation (V) this article uses intend to be such that no cell is mapped onto itself (ie: (i,j) <-> (H[i],V[j]) H[i] <> i or V[j] <> j for i,j = 0 .. m-1)
self inverse permutation factor: F(1) = 1 F(2) = 1 + (²₂) = 2 F(3) = 1 + (³₂) = 4 F(4) = 3 F(2) + F(3) = 10 F(m) = (m-1) F(m-2) + F(m-1)	Permutation which is it's own inverse ie: P[P[i]] = i
	The first permutation element is involved in m-1 swaps, the other elements for obvious reason obey the order m-2 formula. When the first element is not involved the m-1 other elements are subject to the order m-1 argumentation
	(0)	(0,1,2,3) (0,1,3,2) (0,2,1,3) (0,3,1,2) (1,0,2,3) (1,0,3,2) (2,1,0,3) (2,3,0,1) (3,1,2,0) (3,2,1,0)
	(0,1) (1,0)
	(0,1,2) (0,2,1) (1,0,2) (2,1,0)
Diagonal symmetric patterns	pattern symmmetric in main diagonal
Diagonal symmetric patterns	With permutations such that P[i] <> i, the permutation can be used in both directions (Dudeney pattern I, II and III are of this type (see Dudeney article))
1-agonal invariant paterns	pattern swapping elements within 1 1-agonal
1-agonal invariant paterns	With permutations such that P[i] <> i, the permutation can be used in one of the directions while the other direction the natural order is maintained (ie combined with P[i] == i) (Dudeney patterns IV / X are of this type (see Dudeney article))
central 1-agonal symmetric patterns	patttern symmetric in central 1-agonal
central 1-agonal symmetric patterns	With symmetric permutations (ie: P[i] + P[m-1-i] = m-1), used in one of the directions the pattern is symmetric in the central perpendicuar direction (all Dudeney patterns are of this type (see Dudeney article))
Theoretical basis (p-tuple permutations) (preliminairy)
p-tuple permutation	permutation which whe applied p times results in the identity
p-tuple permutation	permutations like (1,2,3,0) applied 4 time result in the identity: (1,2,3,0)(1,2,3,0)(1,2,3,0)(1,2,3,0) = (1,2,3,0)(1,2,3,0)(2,3,0,1) = (1,2,3,0)(3,0,1,2) = (0,1,2,3) These kind of permutations applied in the same manner in 1-agonal directions as the self-inverse permutations lays relations between (p=4) cells
Theoretical basis (pattern desccription with p-tuple permutations) (preliminairy)
patterns	use of multipe p-tuple permutations can describe all kinds of patterns
patterns	Once expertise is obtained (by programmatic experimentation) the usefullness of this avenue can be ascertained (pe {(1,2,3,4,0),(4,3,1,0)} pair in both direction define an order 5 X pattern)