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Introduction The Order-16 Patterns Some example squares

Introduction

With order-16 quadrant magic squares we open up many more possibilities. Instead of 2 possible patterns for order-8 and 10 possible symmetrical patterns for order 12, we now have 52 symmetrical patterns (as predicted by Aale de Winkel).

As with the order-12 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 non-symmetrical 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 Order-16 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 non-symmetrical 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 order-16 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 order-16 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. 
S1 = 3056, S2 = 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 order-8 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 non-symmetrical patterns such as
A, H, L, and T.

This page was originally posted July 23, 2011
It was last updated July 24, 2011
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz