The Magic Encyclopedia ™

The Quadrant magic (feature)
(by Aale de Winkel)

A square is said to be "Quadrant magic" with respect to a certain pattern, if that pattern defined by k cells in a p by q rectangle is repeated 4 times within the square and each of the four copies of the pattern is summing to the same sun which is: k (m2 + 1) / 2.
The original idea of the quadrant magic feature came from <Harvey Heinz> for the orders of type m = 4k+1 with totally symmetric patterns of m cells in the 2k+1 by 2k+1 cornered subsquares summing to the magic sum of the order m square investigated in: Quadrant Magic The idea was rdefined in the other types of orders later on by this author upon suggestion by Harvey
Recent finds by <Walter Trump> in order 7 center symmetric pandiagonal squares of quite interesting patterns lead me to augment the definition further to the definition above. In his "Harvey Heinz" square he even found real Quadrant Bimagic "H" patterns (ie. The H patterns normal and squared sums were the same in all quadrants)
The below defines the quadrant magic feature and connected issues. In lite of recent altered viewpoints and research this article tries to keep up with curent practise, especially the counting arguments need much work with resect to that.

The Quadrant magic (feature)
The following defined the possibe symmetry types of a pattern
Quadrant totally symmetric pattern of m cells
Total totally symmetric pattern (any other number of cells)
Windmill rotation symmetric pattern
Lines pattern symmetric in both central lines
Diagonals pattern symmetric in both diagonals
Center pattern symmetric in center
Main diagonal pattern symmetric in main diagonal
sub diagonal pattern symmetric in sub diagonal
Horizontal pattern symmetric in central horizontal line
Vertical pattern symmetric in central vertical line
Not pattern has no symmetry
The following define the copying possibilities of a pattern
The "Placement Policy"
mirrored pattern is mirrored
copyed pattern is copied
rotated pattern is rotated
order dependent quadrant definitions
orders quadrants 1st kind
total patterns
2nd kind
total patterns
4k-1 2k-1 by 2k-1 subsquare
free central line
from adjoining quadrants
((2k-1)^24k-1) ((2k-1)^22k-1)
2k by 2k subsquare
common central line
to two adjoining quadrants
((2k)^24k-1) ((2k)^22k)
4k 2k by 2k subsquare
adjoining quadrants
((2k)^24k) ((2k)^22k)
4k+1 2k+1 by 2k+1 subsquare
common central line
to two adjoining quadrants
((2k+1)^24k+1) ((2k+1)^22k+1)
2(2k+1) 2k+1 by 2k+1 subsquare
adjoining quadrants
((2k+1)^22(2k+1)) ((2k+1)^22k+1)
pattern counting formulae
4k-1 quadrant none
windmill none
lines/diags j=1k-1 ((2k-2)j)((k-1)^2k-j)
line/diag. j=0k(2k-12k-1-2j)((k-1)(2k-1)k+j)
center (2k(k-1)k-1)
4k quadrant j=0k div 2 (kk-2j)(k(k-1)/2j)
windmill (k^2k)
lines (k^2k)
diagonals j=0k (2k2k-2j)(k(k-1)j)
line (2k^22k)
diagonal j=0k (2k2k-2j)(k(2k-1)k+j)
center (2k^22k)
4k+1 quadrant j=0k div 2(k(k-1) div 2j)(2kk-2j)
windmill (k(k+1)k)
lines/diags j=0k (2k2j)(k*kk-j)
line/diag. j=0k(2k+12j+1)(k(2k+1)2k-j)
center (2k(k+1)2k)
2(2k+1) quadrant none
windmill none
lines/diags j=0k (2k2j+2)(k*kk-j)
line/diag. j=0k(2k+12j)(k(2k+1)2k+1-j)
center (2k(k+1)2k+1)