Order17 Quadrant Magic Squares


Introduction 
A brief introduction to some of the 253 quadrant magic arrays of order17. 
1way quadrant magic 
This pandiagonal magic square is p082 (sring quadrant magic). 
2way quadrant magic 
This pandiagonal square is both p112 and p153 magic. 
3way quadrant magic 
This pandiagonal square is p209, p213 and p216 magic. 
4way quadrant magic 
This pandiagonal square is p118, p209, p215 and p241 magic. 
5way quadrant magic 
This pandiagonal square is p001, p085, p118, p178 and p209 magic. 
6way quadrant magic 
This pandiagonal square is p070, p153, p172, p178, p215 and p248 magic. 
Regular 1way quadrant magic 
This regular square is p241 magic but contains at least 1 of each of 15 arrays. 
Regular 3way quadrant magic 
This regular square is p001 (plus), p082 (sring) and p085 magic. 
Regular 4way quadrant magic 
This regular square is p153, p172, p178 and p248 magic. 
Summary and questions 
The following 15 patterns are a sample from the set of 253 quadrant magic arrays of order17.
This is the highest of 10 types of magic arrays as determined by degree of symmetry, and is the only type used to form quadrant magic squares.
For order17, the quadrants are 9 x 9 cells. The magic array has 16 cell plus
the quadrant's center cell, that together must sum to the magic constant.
Remember, to be quadrant magic, a magic square must have 4 identical
arrays in the 4 quadrants.
The plusmagic (p001) quadrant magic square is the only type that can transform
into an order17 isolike magic star.
Quadrant magic squares and arrays were investigated
jointly by Aale de Winkel and myself during the period of May to
September, 1999. Please visit his site at
http://www.magichypercubes.com/ 
The sets of 4 identical arrays that occupy the 4 quadrants of the example quadrant magic squares  
Type  LP (Latin Prescription)  p 001 
p 070 
p 082 
p 085 
p 112 
p 118 
p 153 
p 172 
p 178 
p 209 
p 213 
p 215 
p 216 
p 241 
p 248 
# of arrays  
Sqr.  Quad  
Pan  (2,3,0)(2,11,0)  .  .  @  .  .  .  .  .  .  .  .  .  .  .  .  11  7 
Pan  (1,8,0)(2,1,0)  .  .  .  . 
@ 
. 
@ 
.  .  .  .  .  .  .  .  8  5 
Pan  (1,10,0)(12,15,0)  .  .  .  .  .  .  .  .  .  @  @  . 
@ 
.  .  14  8 
Pan  (7,11,0)(11,7,0)  .  .  .  .  .  @  .  .  . 
@ 
. 
@ 
. 
@ 
.  12  10 
Pan  (5,14,0)(14,5,0) 
@ 
..  . 
@ 
.  @  .  .  @  X  .  .  .  .  .  13  9 
Pan  (10,12,0)(12,10,0)  . 
@ 
.  .  .  .  X  @  @  .  . 
X 
.  . 
@ 
15  15 
.  .  .  .  .  
Reg  (7,11,0)(8,8,0)  .  .  .  .  .  .  .  .  .  .  .  .  . 
@ 
.  15  15 
Reg  (8,8,0)(14,1,0) 
@ 
.  @  @  .  .  .  .  .  .  .  .  .  .  .  14  10 
Reg  (8,8,0)(12,10,0)  .  .  .  .  .  .  @  @  @  .  .  .  .  . 
@ 
12  8 
..@  = this array shown in diagram  ..X  = this array is present in all quadrants but not shown in diagram 
The last two columns show the total number of magic arrays that are present at least once in the square, and the number in the most populated quadrant.
This square is P082 (sring) quadrant magic but contains 11different quadrant magic arrays, 7 in the upper right quadrant. This and the following quadrant magic squares show the center cell of each quadrant green. This cell is common to all the magic arrays in that quadrant of the square. LP is (2, 3, 0)(2, 11, 0).

This square is P112 (diam) and p153 quadrant magic.
It contains 8 different quadrant magic arrays, 5 in the upper left
quadrant. LP is (1, 8, 0)(2, 1, 0). 
This square is p209, p213 and, p216 quadrant magic.
It contains 14 different quadrant magic arrays, 8 in the upper right
quadrant. It is LP (1, 10, 0)(12, 15, 0). Here blue indicates p209, yellow p213 (lring) and violet p216. Showing multiple magic arrays in the
same diagram can be confusing when the arrays share cells in the
common row or common column. I will use green to indicate all cells
that are common to 2 adjacent arrays (as well as the central cell of
each quadrant). 
This square is p118, p209, p215 and, p241 quadrant
magic. It contains 12 different quadrant magic arrays, 10 in the
upper right quadrant. This is LP (7 ,11, 0)(11, 7, 0). P118 is blue, p209 is yellow, p215 is violet and p241 is red.. 
This square is p001 (plus), p085, p118, p178 and
p209 quadrant magic. It contains 12 different quadrant magic arrays,
10 in the upper left quadrant. P001 (plus) is blue, p085 is yellow, p118 is violet and p178 is red. Not shown is p209. 
This square is p070, p153, p172, p178, p215 and,
p248 quadrant magic. It contains all 15 different quadrant magic
arrays and all 15 in the upper left quadrant.
This is LP (10, 12, 0)(12, 10, 0). A reminder; 
This square has only the p241 quadrant magic array
in all 4 quadrants. However, it contains at least 1 copy of all 15 different quadrant magic arrays, and all 15 appear in the upper left quadrant. It is LP (7, 11, 0)(8, 8, 0) It, and the following 2 examples are regular (not pandiagonal) magic squares, 
This square is a regular p001, p 082 and p085
quadrant magic square. It contains 14 different quadrant magic
arrays with 10 in the upper right quadrant. It is LP (8, 8, 0)(14, 1, 0). Blue is p001 (plus), yellow is p082 (sring), violet is p085, green is quadrant centers and cells common to 2 adjacent arrays. Notice that all regular (not pandiagonal) quadrant magic squares shown here have 1 of the 2 LP vectors as (8, 8, 0).

Some reminders:
All magic arrays shown appear in all 4 quadrants of the magic square.
Some patterns appear to be smaller then others, but all are considered to
occupy the entire quadrant of the magic square.
The central row and central column of the magic square is common to the 2
adjacent quadrants.
This square is a regular p153, p172, p178 and p248
quadrant magic square. It contains 12 different quadrant magic
arrays with 8 in the upper left and upper right quadrants.
This is LP (8, 8, 0)(12, 10, 0). 
Quadrant magic arrays are the most symmetrical of 10 classes of magic arrays. It is the only type of magic array considered on these pages, and the only type (by definition) that can form quadrant magic squares.
The magic squares shown on this page were found by searching for only
15 of the 253 order17 quadrant magic arrays. If a search is conducted for all
253 quadrant magic arrays:
What is the largest
possible number of 4 identical arrays to appear in the same magic square?
What is the largest number that appears at least once in the same magic
square?
What is the largest number that appears in the same quadrant of the same
magic square?
I have shown at least 1 example of a quadrant magic square formed
from each of the 15 arrays searched for.
Can magic squares be
formed using each of the 253 quadrant magic arrays?


This page was originally posted August 1999 