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