The Magic Encyclopedia Ptuple 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 cellcoordinates 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 .. m1) 

self inverse permutation factor: F(1) = 1 F(2) = 1 + (^{2}_{2}) = 2 F(3) = 1 + (^{3}_{2}) = 4 F(4) = 3 F(2) + F(3) = 10 F(m) = (m1) F(m2) + F(m1) 
Permutation which is it's own inverse ie: P[P[i]] = i  
The first permutation element is involved in m1 swaps, the other elements for obvious reason obey the order m2 formula. When the first element is not involved the m1 other elements are subject to the order m1 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)) 

1agonal invariant paterns  pattern swapping elements within 1 1agonal  
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 1agonal symmetric patterns  patttern symmetric in central 1agonal  
With symmetric permutations (ie: P[i] + P[m1i] = m1), 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 (ptuple permutations) (preliminairy) 

ptuple 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 1agonal directions as the selfinverse permutations lays relations between (p=4) cells 

Theoretical basis (pattern desccription with ptuple permutations) (preliminairy) 

patterns  use of multipe ptuple 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) 