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=1∑k-1 ((2k-2)j)((k-1)^2k-j)
	line/diag.	j=0∑k(2k-12k-1-2j)((k-1)(2k-1)k+j)
	center	(2k(k-1)k-1)
4k	quadrant	j=0∑k div 2 (kk-2j)(k(k-1)/2j)
	windmill	(k^2k)
	lines	(k^2k)
	diagonals	j=0∑k (2k2k-2j)(k(k-1)j)
	line	(2k^22k)
	diagonal	j=0∑k (2k2k-2j)(k(2k-1)k+j)
	center	(2k^22k)
4k+1	quadrant	j=0∑k div 2(k(k-1) div 2j)(2kk-2j)
	windmill	(k(k+1)k)
	lines/diags	j=0∑k (2k2j)(k*kk-j)
	line/diag.	j=0∑k(2k+12j+1)(k(2k+1)2k-j)
	center	(2k(k+1)2k)
2(2k+1)	quadrant	none
	windmill	none
	lines/diags	j=0∑k (2k2j+2)(k*kk-j)
	line/diag.	j=0∑k(2k+12j)(k(2k+1)2k+1-j)
	center	(2k(k+1)2k+1)