This page became necessary when the material on order4 transformations accumulated beyond my wildest expectations. I started with a page called ‘Transformations and Patterns’. I soon had to start another one called "More Order4 Transformations" which also grew too fast. Hopefully this one will be sufficient to hold any remaining material!
However, my hope is that in reading this material, you will say "Ahha, but how
about …".
I am well aware that there is still much to discover about order4 magic squares and
methods of transforming one to another. I look forward to comments, constructive criticism
and hearing of new discoveries.
Hey, I just thought, how about complementing the LSD of the octal representation, or how
about ...
Binary Digit Swap 
6 transitions that work for all groups I to VIP by exchanging some
digits. 
Complementing binary digits 
6 transitions that work for all groups I to VIP by complementing some
digits. 
Summary 
A table lists 48 transformations that work on all magic squares of at
least 1 Dudeney group, showing characteristics. 30 work on ALL groups I to VIP. 
Addendum Mar. 12, 2002 
Some slightly differing results from Holger Danielsson. 
Intro to Order4 Transforms. 
Back to the introduction page to this subject. (Also up arrow above and at
end). 
More Order4 Transforms 
Page 2 of 4 pages on this subject. 
Fellows Transformations. 
His base4 digit manipulation transformations. Also a 4 magic square loop. 
The investigation of the following transformations was motivated by reviewing the work
Ralph Fellows is doing with transformations involving manipulation with the digits of the
magic square numbers.
He has developed several transformations involving base 4 representation. This gave me the
idea to try the same with base 2 representation.
While he has concentrated on developing transformations that may be used with any order, I
choose to restrict my investigations to transformations that may work only with order4
magic squares. Of course the binary number system is ideal for representing order4
numbers because 4 binary digits exactly covers the decimal range 0 to 15.
The numbers 0 to 15 in a magic square may be represented by the binary numbers 0 to 1111.
Then if we swap a pair of binary digits and convert the resulting 4 digit number back to decimal, a new magic square may be obtained.
Call the digits a, b, c, and d starting from the left. This first procedure involves swapping digits a and d.
Original Dec 0 to 15 change to base 2 Swap a and d Dec 0 to 15 Dec 1 to 16 112 III 203 III 01 08 12 13 00 07 11 12 0000 0111 1011 1100 0000 1110 1011 0101 00 14 11 05 01 15 12 06 14 11 07 02 13 10 06 01 1101 1010 0110 0001 1101 0011 0110 1000 13 03 06 08 14 04 07 09 15 10 06 03 14 09 05 02 1110 1001 0101 0010 0111 1001 1100 0010 07 09 12 02 08 10 13 03 04 05 09 16 03 04 08 15 0011 0100 1000 1111 1010 0100 0001 1111 10 04 01 15 11 05 02 16
This procedure is the first entry in the following table which shows 5 other binary digit interchanges that also produce magic squares of the same group as the original.
Decimal 0 – 15  0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
= Binary  0000 
0001 
0010 
0011 
0100 
0101 
0110 
0111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
Exchange a  d  0000 
1000 
0010 
1010 
0100 
1100 
0110 
1110 
0001 
1001 
0011 
1011 
0101 
1101 
0111 
1111 
Substitute number  0 
8 
2 
10 
4 
12 
6 
14 
1 
9 
3 
11 
5 
13 
7 
15 
Exchange b  c  0000 
0001 
0100 
0101 
0010 
0011 
0110 
0111 
1000 
1001 
1100 
1101 
1010 
1011 
1110 
1111 
Substitute number  0 
1 
4 
5 
2 
3 
6 
7 
8 
9 
12 
13 
10 
11 
14 
15 
Exchange a  c  0000 
0001 
1000 
1001 
0100 
0101 
1100 
1101 
0010 
0011 
1010 
1011 
0110 
0111 
1110 
1111 
Substitute number  0 
1 
8 
9 
4 
5 
12 
13 
2 
3 
10 
11 
6 
7 
14 
15 
Exchange b  d  0000 
0100 
0010 
0110 
0001 
0101 
0011 
0111 
1000 
1100 
1010 
1110 
1001 
1101 
1011 
1111 
Substitute number  0 
4 
2 
6 
1 
5 
3 
7 
8 
12 
10 
14 
9 
13 
11 
15 
Exchange ac, bd  0000 
0100 
1000 
1100 
0001 
0101 
1001 
1101 
0010 
0110 
1010 
1110 
0011 
0111 
1011 
1111 
Substitute number  0 
4 
8 
12 
1 
5 
9 
13 
2 
6 
10 
14 
3 
7 
11 
15 
Exchange ad, bc  0000 
1000 
0100 
1100 
0010 
1010 
0110 
1110 
0001 
1001 
0101 
1101 
0011 
1011 
0111 
1111 
Substitute number  0 
8 
4 
12 
2 
10 
6 
14 
1 
9 
5 
13 
3 
11 
7 
15 
Renumber the original magic square using integers 0 to 15 then look up the corresponding number. Increase each number in the magic square by 1 to obtain a new magic square with integers 1 to 16
All these transformations are reversible. Apply the same transformations the second time and the original magic square is obtained.
Exchanging the first two binary digits and then exchanging the last two digits result in no successful transformations.
Results of above transformations
I 
II 
III 
IV 
V 
VIP 
VIS 
VII 
VIII 
IX 
X 
XI 
XII 

Exchange a & d  all? 
all? 
all? 
all? 
all? 
all? 
some 
some 
none? 
some 
some 
none 
none 
Exchange b & c  all? 
all? 
all? 
all? 
all? 
all? 
some 
none? 
none? 
none? 
none? 
none 
none 
Exchange a & c  all? 
all? 
all? 
all? 
all? 
all? 
some 
none? 
none? 
none? 
none? 
none 
none 
Exchange b & d  all? 
all? 
all? 
all? 
all? 
all? 
some 
none? 
none? 
none? 
none? 
none 
none 
Exchange ac, bd  all? 
all? 
all? 
all? 
all? 
all? 
some 
some 
some 
some 
some 
1 only 
none 
Exchange ad, bc  all? 
all? 
all? 
all? 
all? 
all? 
some 
none? 
none? 
none? 
none? 
none 
none 
All magic squares I tested of groups I to VIP transformed successfully, but in each
case, the resulting magic square was different
In all cases the resulting magic square belonged to the same group as the original one.
The question marks in the above table indicate that I have not tested all magic squares in
that group so there could still be an exception.
Shortcut for above transformations
Simply substitute the following numbers for the numbers of the original magic square to obtain the transformed one.
Decimal 1 to 16  1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
Exchange a & d  1 
9 
3 
11 
5 
13 
7 
15 
2 
10 
4 
12 
6 
14 
8 
16 
Exchange b & c  1 
2 
5 
6 
3 
4 
7 
8 
9 
10 
13 
14 
11 
12 
15 
16 
Exchange a & c  1 
2 
9 
10 
5 
6 
13 
14 
3 
4 
11 
12 
7 
8 
15 
16 
Exchange b & d  1 
5 
3 
7 
2 
6 
4 
8 
9 
13 
11 
15 
10 
14 
12 
16 
Exchange ac, bd  1 
5 
9 
13 
2 
6 
10 
14 
3 
7 
11 
15 
4 
8 
12 
16 
Exchange ad, bc  1 
9 
5 
13 
3 
11 
7 
15 
2 
10 
6 
14 
4 
12 
8 
16 
An Example using the same original magic square for 6 transformations.
Original Swap a<>d Swap b<>c Swap a<>c Swap b<>d Swap ac,bd Swap ad,bc 32 V 165 V 66 V 114 V 97 V 103 V 173 V 01 04 16 13 01 11 16 06 01 06 16 11 01 10 16 07 01 01 16 10 01 13 16 04 01 13 16 04 14 15 03 02 14 08 03 09 12 15 05 02 08 15 09 02 14 12 03 05 08 12 09 05 12 08 05 09 07 06 10 11 07 13 10 04 07 04 10 13 13 06 04 11 04 06 13 11 10 06 07 11 07 11 10 06 12 09 05 08 12 02 05 15 14 09 03 08 12 03 05 14 15 09 02 08 15 03 02 14 14 02 03 15
Complementing individual digits of the binary representation of the magic square numbers also result in successful transformations.
Again we will identify the digits as a, b, c, d starting from the left hand (MSD) digit.
Original Dec 0 to 15 change to base 2 Swap a and d Dec 0 to 15 Dec 1 to 16 112 III 789 III 01 08 12 13 00 07 11 12 0000 0111 1011 1100 1000 1111 0011 0100 08 15 03 04 09 16 04 05 14 11 07 02 13 10 06 01 1101 1010 0110 0001 0101 0010 1110 1001 05 02 14 09 06 03 15 10 15 10 06 03 14 09 05 02 1110 1001 0101 0010 0110 0001 1101 1010 06 01 13 10 07 02 14 11 04 05 09 16 03 04 08 15 0011 0100 1000 1111 1011 1100 0000 0111 11 12 00 07 12 13 01 08
This procedure is the first entry in the following table which shows 5 other binary
digit complement transformations.
Here I show the original magic square number and the decimal number to substitute for it
to obtain the new magic square.
These numbers were found by working with the binary representation of the decimal numbers
0 to 15, (similar to the above example).
All these transformations produce different magic squares but in each case the new square
belongs to the same group as the original.
Complementing a and b or c and d is the same as complementing the MSD or the LSD of the
base 4 representation (Fellows).
Not yet tested. Complementing 3 of the 4 binary digits (i.e. a, b, c or a, b, d, or a, c,
d or b, c, d).
Decimal 1 – 16  1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
Complement a (MSD)  9 
10 
11 
12 
13 
14 
15 
16 
1 
2 
3 
4 
5 
6 
7 
8 
Complement b  5 
6 
7 
8 
1 
2 
3 
4 
13 
14 
15 
16 
9 
10 
11 
12 
Complement c  3 
4 
1 
2 
7 
8 
5 
6 
11 
12 
9 
10 
15 
16 
13 
14 
Complement d (LSD)  2 
1 
4 
3 
6 
5 
8 
7 
10 
9 
12 
11 
14 
13 
16 
15 
Complement a and c  11 
12 
9 
10 
15 
16 
13 
14 
3 
4 
1 
2 
7 
8 
5 
6 
Complement b and d  6 
5 
8 
7 
2 
1 
4 
3 
14 
13 
16 
15 
10 
9 
12 
11 
Results of above transformations
I 
II 
III 
IV 
V 
VIP 
VIS 
VII 
VIII 
IX 
X 
XI 
XII 

Complement a (MSD)  all? 
all? 
all? 
all? 
all? 
all? 
some 
some 
some  some  some 
some 
some 
Complement b  all? 
all? 
all? 
all? 
all? 
all? 
some 
some 
some  some  some 
some 
none 
Complement c  all? 
all? 
all? 
all? 
all? 
all? 
some 
none? 
some  some  none? 
none 
none 
Complement d (LSD)  all? 
all? 
all? 
all? 
all? 
all? 
some 
some 
some  some  none? 
some 
some 
Complement a and c  all? 
all? 
all? 
all? 
all? 
all? 
some 
none? 
none?  none?  none? 
none 
none 
Complement b and d  all? 
all? 
all? 
all? 
all? 
all? 
some 
none? 
none?  none?  none? 
none 
none 
All magic squares I tested of groups I to VIP transformed successfully, but in each
case, the resulting magic square was different (but duplicates within each group. see
below).
In all cases the resulting magic square belonged to the same group as the original one.
The question marks in the above table indicate that I have not tested all magic squares in
that group so there could still be an exception.
An Example using the same original magic square for 6 transformations.
Original comple. a comple. b comple. c comple. d comple. a,c comple. b,d 32 V 577 V 577 V 425 V 228 V 228 V 425 V 01 04 16 13 09 12 08 05 05 08 12 09 03 02 14 15 02 03 15 14 11 10 06 07 06 07 11 10 14 15 03 02 06 07 11 10 10 11 07 06 16 13 01 04 13 16 04 01 08 05 09 12 09 12 08 05 07 06 10 11 15 14 02 03 03 02 14 15 05 08 12 09 08 05 09 12 13 16 04 01 04 01 13 16 12 09 05 08 04 01 13 16 16 13 01 04 16 11 07 06 11 10 06 07 02 03 15 14 15 14 02 03
Notice that there are only 3 new magic squares with 2 versions of each.
I was shocked when I discovered this and suspected I had made a mistake somewhere.
However, on further investigation, I found this was general for all groups I to VIP. In
each case that I investigated (except 2). I found 3 sets of two new squares. There seems
to be no order as to how these sets are arranged. I present 2 examples from each group.
Original magic square and group  Complement 
Complement 
Complement 
Complement 
Complement 
Complement 
102  I  828 
785 
279 
279 
785 
828 
116  I  647 
364 
485 
304 
304 
116 
21  II  591 
213 
445 
213 
213 
21 
27  II  583 
421 
421 
233 
233 
583 
112  III  789 
789 
289 
289 
834 
834 
113  III  790 
790 
290 
290 
835 
835 
24  IV  216 
589 
443 
216 
589 
443 
735  IV  191 
399 
399 
572 
572 
191 
32  V  577 
577 
425 
228 
228 
425 
173  V  853 
798 
362 
362 
798 
853 
16 VIP  435 
224 
435 
224 
16 
16 
638 VIP  298 
490 
298 
490 
638 
638 
In each case the magic square generated by the transformation is a member of the same
group as the originating magic square.
I have indicated the pairs in a group by colors (one II has a triplet).
#116 and # 21 both have generated magic squares that are not paired up. They both have
disguised versions of themselves.
However, # 638 produced two disguised versions of itself!
I’d say this is a pretty mixed up situation. Very unlike the orderly results of most
of the transitions.
Transformation  I 
II 
III 
IV 
V 
VIP 
VIS 
VII 
VIII 
IX 
X 
XI 
XII 

Complement each number  1 
2 
3 
4 
5 
6P 
6S 
7 
8 
9 
10 
11 
12 

1  Swap rows 1 and 2   
2 
 
 
 
 
 
 
 
 
 
 
 
1  Swap columns 1 and 2   
2 
 
 
 
 
 
 
 
 
 
 
 
1  Swap rows and columns 1 and 2  3 
2 
1 
4 
6P 
5 
 
 
 
 
 
 
 
2  Swap rows 1 and 3  1 
 
 
 
 
 
 
 
 
 
 
 
 
2  Swap columns 1 and 3  1 
 
 
 
 
 
 
 
 
 
 
 
 
2  Swap rows and columns 1 and 3  1 
3 
2 
6P 
5 
4 
 
 
 
 
 
 
 
3  Swap rows 1 and 4   
 
3 
 
 
 
 
 
 
 
 
 
 
3  Swap columns 1 and 4   
 
3 
 
 
 
 
 
 
 
 
 
 
3  Swap rows and columns 1 and 4  2 
1 
3 
5 
4 
6P 
6S 
10 
9 
8 
7 
12 
11 
3  Swap rows 2 and 3   
 
3 
 
 
 
 
 
 
 
 
 
 
3  Swap columns2 and 3   
 
3 
 
 
 
 
 
 
 
 
 
 
3  Swap rows and columns 2 and 3  2 
1 
3 
5 
4 
6P 
6S 
10 
9 
8 
7 
12 
11 
2  Swap rows 2 and 4  1 
 
 
 
 
 
 
 
 
 
 
 
 
2  Swap columns2 and 4  1 
 
 
 
 
 
 
 
 
 
 
 
 
2  Swap rows and columns 2 and 4  1 
3 
2 
6P 
5 
4 
 
 
 
 
 
 

1  Swap rows 3 and 4   
2 
 
 
 
 
 
 
 
 
 
 
 
1  Swap columns 3 and 4   
2 
 
 
 
 
 
 
 
 
 
 
 
1  Swap rows and columns 3 and 4  3 
2 
1 
4 
6P 
5 
 
 
 
 
 
 
 
Change row & col. order to 1342  2 
3 
1 
6P 
4 
5 
 
 
 
 
 
 
 

Change row & col. order to 1423  3 
1 
2 
5 
6P 
4 
 
 
 
 
 
 
 

4  Change row order to 2143  1 
2 
3 
4 
5 
6P 
 
 
 
 
 
 
 
4  Change column order to 2143  1 
2 
3 
4 
5 
6P 
 
 
 
 
 
 
 
4  Change row & column order to 2143  1 
2 
3 
4 
5 
6P 
6S 
9 
10 
7 
8 
11 
12 
Change row order to 3142   
 
3 
 
 
 
 
 
 
 
 
 
 

Change col. order to 3142   
 
3 
 
 
 
 
 
 
 
 
 
 

Change row & col. order to 3142  2 
1 
3 
5 
4 
6P 
6S 
8 
7 
10 
9 
12 
11 

5  Move rows and/or col. to opposite side  1 
 
 
 
 
 
 
 
 
 
 
 
 
Move quadrants clockwise   
 
3 
 
 
 
 
 
 
 
 
 
 

Move quadrants counterclockwise   
 
3 
 
 
 
 
 
 
 
 
 
 

6  Convert quadrants to rows  3 
 
1 
 
 
 
 
 
 
 
 
 
 
7  Change diagonals to rows  5 
4 
6P 
4/2 
1/5 
36S 
6P* 
 
 
 
 
 
 
8  Exchange binary digits a and d  1 
2 
3 
4 
5 
6P 
some 
some 
 
some 
some 
none 
none 
8  Exchange binary digits b and c  1 
2 
3 
4 
5 
6P 
some 
 
 
 
 
none 
none 
8  Exchange binary digits a and c  1 
2 
3 
4 
5 
6P 
some 
 
 
 
 
none 
none 
8  Exchange binary digits b and d  1 
2 
3 
4 
5 
6P 
some 
 
 
 
 
none 
none 
8  Exchange binary digits a and c, b and d  1 
2 
3 
4 
5 
6P 
some 
some 
some 
some 
some 
some 
none 
8  Exchange binary digits a and d, b and c  1 
2 
3 
4 
5 
6P 
some 
 
 
 
 
none 
none 
Complement binary digit a (MSD)  1 
2 
3 
4 
5 
6P 
some 
some 
some 
some 
some 
some 
some 

Complement binary digit b  1 
2 
3 
4 
5 
6P 
some 
some 
some 
some 
some 
some 
none 

Complement binary digit c  1 
2 
3 
4 
5 
6P 
some 
 
some 
some 
 
none 
none 

Complement binary digit d (LSD)  1 
2 
3 
4 
5 
6P 
some 
some 
some 
some 
some 
some 
some 

9  Complement binary digit a and c  1 
2 
3 
4 
5 
6P 
some 
 
 
 
 
none 
none 
Complement binary digit b and d  1 
2 
3 
4 
5 
6P 
some 
 
 
 
 
none 
none 

10  Base 4 digit swap (Fellows)  1 
2 
3 
4 
5 
6P 
some 
some 
some 
some 
some 
none 
none 
10  Complement Base 4 LSD (Fellows)  1 
2 
3 
4 
5 
6P 
some 
some 
some 
some 
some 
none 
none 
10  Complement Base 4 MSD (Fellows)  1 
2 
3 
4 
5 
6P 
some 
some 
some 
some 
some 
none 
none 
Congruent modulo 8 (SaintPierre)  1  2  3  4  5  6P    7  8         
Notes:
Holger Danielsson, summarized two months of investigations into this subject with the
following document, emailed to me Feb. 26, 2002
I have edited it slightly for brevity, and offer it here with no further
comment. He no longer seems to have a web site (Sept./09).
I have not confirmed his findings, but this illustrates how complex this subject is.
Swap rows OR columns only
· works
for all 208 squares of the groups 1..5 and the semipandiagonal squares of group 6a
· don’t
work for groups 6b...12
in table 1 you can see, how many different squares are created
But surprisingly enough, there are differences depending on what s
squares are used (normalized squares, squares arranged like the Dudeney pattern, and my
squares built with the additions tables of BensonJacoby)
a) transformation of group 2
squares will create all 48 squares in this group, which doesn’t hold for the other
groups
b) transformations of groups
2, 4, 5, 6a will create all squares, when using squares, which are arranged like the
Dudeney pattern.
c) and most surprising: the
squares which i built from the additions tables of BensonJacoby will create all squares
of the group. This is true for every group.
Swap rows OR columns
44 (68) means that 68 squares are transformed to this group, where only 44 of them are different. If only one number shown, all transformed squares are different. 
Diagonals to rows

Change diagonals to rows
This is the part with errors as you can see in table 2:
· the
transformed squares are in the same group and you will get all squares for group 1..3
· but
this is not true for groups 4, 5 and 6a. As you can see for example, there are 28 of the
normalized squares which will stay in group
5,
but 68 of them are transformed to
group 1, where 44 of them are really different.
· you
are right, when you say that the destination group is determinedby the orientation of the
complement pairs (squares with horizontal pairs
go to group 1, squares with vertical
pairs will stay in group 5)
· but
it is false that 48 will stay and 48 will
change the group (see the results above). Where do you have these counts from?
· and
still surprising again: using my BensonJacoby squares or the squares arranged like the
Dudeney pattern all squares will stay in their
groups and also create all the squares in
the group.
Holger Danielsson Feb. 26, 2002
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Last updated
September 11, 2009
Harvey Heinz harveyheinz@shaw.ca
Copyright © 2000 by Harvey D. Heinz