Index
Introduction
With order16
quadrant magic squares we open up many more possibilities. Instead of 2
possible patterns for order8 and 10 possible symmetrical patterns for
order 12, we now have 52 symmetrical patterns (as predicted by Aale de
Winkel).
As with the order12
patterns, I show here just 1/4 of each quadrant pattern. It is the top
left corner, and will need to be reflected appropriately to fill the other
quarters of the 8x8 square quadrant. The quadrant is then placed 4 times
to fill the square, also reflecting as necessary.
Showing just one
quadrant of the quadrant pattern has several distinct advantages made
possible by the symmetry.

Takes just one forth
of the space.

Takes one forth of
the work.

Is easier to see
find more patterns, see the symmetry, and avoid duplicates.
I previously
mentioned the possibility of nonsymmetrical patterns. I have also
included 4 of the many possible. These patterns exist in Compact magic
squares and consist of a combination of 2x2 cell arrays. The pattern may
be placed anywhere in the quadrant.
Four of my 5 example
squares are not compact or associated. Yet all are quadrant magic.
The Order16 Patterns
Some
example squares
Many of the other
examples were compact squares. None of the first 4 examples are compact, but
all are quadrant magic. the fifth square is compact and the test
spreadsheet shows that the nonsymmetrical patterns A. H, L and T all
appear in all 4 quadrants (as we would expect).
Square 1
This 16x16 square is pandiagonal magic
Not compact or associated
Each quadrant is Pandiagonal magic
Not compact, but associated
This is an order16 Quadrant Magic Square
with all 52 patterns appearing in all 4 quadrants.
This is the case with all pandiagonal compact
OR associated magic squares
OR all 4 quadrants compact or associated
arrays
(not necessarily magic squares)
This square is by Dwane Campbell May
17, 2011


Square 2
Like the first square, this
square is
pandiagonal magic
Not compact or associated
Unlike the first square, the quadrants are not
magic squares but are associated
Quadrant pandiagonals and columns
are magic, but not rows
This also is an order16
Quadrant Magic Square
with all 52 patterns appearing in all 4 quadrants.
Actually, the letter H also appears in all 4
quadrants!
This square is by Dwane Campbell May 17,
2011


Square 3
This is a bimagic square.
(see square 4)
When each number is squared (multiplied by itself) the square is still
magic.
S_{1 }= 3056, S_{2} = 351376
This square is pandiagonal magic
Not compact or associated
It is quadrant magic because 14 patterns
appear in all 4 quadrants. The patterns are :
1, 2, 3, 10, 11, 16, 17, 18, 19, 24, 25, 40, 41, 52.
The quadrants are not magic squares
and are not associated
From Gil Lamb's
AUTOBIMAGIC16.xls of Feb. 2002 

Square 4
This is square 3 with all the
numbers squared. S = 351576
This square is simple magic (not
pandiagonal as Square 3)
Not compact or associated
It is quadrant magic because 2
patterns
(P1 and P16) appear in all 4 quadrants
The quadrants are not magic squares
and are not associated
From Gil Lamb's AUTOBIMAGIC16.xls of Feb.
2002 

Square 5
This square is Benjamin
Franklin's popular order 16.
It is pandiagonal and compact. Not associated.
The quadrants have correct
rows and columns,
but the diagonals are incorrect preventing them
from being order8 magic squares.
The quadrants are compact
(because the square is) but also are not associated.
This square is quadrant
magic
because of the compact feature, all 52 patterns
appear in all 4 quadrants.
Also many nonsymmetrical patterns such as
A, H, L, and T. 

