Introduction
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 ...

6 transitions that work for all groups I to VIP by
exchanging some digits. 

6 transitions that work for all groups I to VIP by
complementing some digits. 

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. 

Some slightly differing results from Holger Danielsson. 

Back to the introduction page to this subject. (Also up
arrow above and at end). 

Page 2 of 4 pages on this subject. 

His base4 digit manipulation transformations. Also a 4
magic square loop. 
Binary Digit Swap
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.
Exchange the MSD and the LSD
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
binary digits
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.
Complement a
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
a 
Complement
b 
Complement
c 
Complement
d 
Complement
a and c 
Complement
b and d 
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.
Summary
 The table lists 48 transformations that work on all magic squares in
at least one group. (For example, I do not show the transformations that
add 4 modulo 16 to numbers of a magic square because this transformation
works only on some squares.)
 Also not shown are transformations that return identical magic
squares. For an example, see note 4.
 I show group VI in two columns, the semipandiagonal (P) and the
simple (S).
 All magic squares of groups XI and XII have been tested for all
transformations. I have tested many but not all magic squares of groups
I to X, so the results I show are a conjecture.
 Where a group number is shown as a result of a transformation, I
have found no exception so assume the transformation works for all magic
squares of the group.
 If I indicate the result with the word 'some', I have found
successful solutions and also magic squares that result in nonmagic
squares.
 The '' indicates that no correct solution was found although not
all magic squares of the group were tested...
 'none' indicate all magic squares of that group were tested.

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:
 Swapping 3 and 4 returns different magic squares then swapping 1 and
2.
 Swap 1 and 3 transformation and swap 2 and 4 transformation return
different magic squares.
 Swapping 2 and 3 returns the same magic squares as swapping 1 and 4,
but with different orientation.
 The same as exchanging rows and columns 1 and 2 with 3 and four or
changing order 3412 or 1 and 3, 2 and 4 or swapping kittycorner
quadrants. In each case the result is the same magic square (although
the orientation may be different.
 Moving 1 row at a time and then cycling through moving the columns
result in a loop of 16 pandiagonal magic squares.
 Converting the quadrants of a group III (associated) magic square to
rows will form a group I (pandiagonal) magic square.
By starting with a different quadrant each time, and cycling through the
4 positions of the quadrant a loop of 16 pandiagonal magic squares are
created.
The transformation works in reverse to form 16 associated magic squares
from 1 pandiagonal.
 Using main and short diagonal pairs. Groups IV to VIP go to 2 other
groups depending on orientation. 48 of VIS go to VIP.
 a is Most Significant Digit, d is Least Significant
Digit.
All transformations involving binary digits result in magic squares
belonging to same group as original.
 Complementing a and b (or c and d) is the same as complementing the
MSD (or LSD) of the base 4 number.
 These 3 Base 4 transformations all return different magic squares.
See these on my Fellows page.
Addendum Mar.
12, 2002
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
from group 

transformed squares 
to group 
Benson 
Dudeney 
normalized 
1 
1 
48 
44 (48) 
44 (48) 
2 
2 
48 
48 
48 
3 
3 
48 
40 (48) 
40 (48) 
4 
4 
96 
96 
84 (96) 
5 
5 
96 
96 
92 (96) 
6a 
6a 
96 
96 
92 (96) 
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
from group 

transformed
squares 
to group 
Benson 
Dudeney 
normalized 
1 
1 
1 
1 
44 (48) 
2 
2 
2 
2 
48 
3 
3 
3 
3 
40 (48) 
4 
4 
96 
96 
84 (84) 

2 


12 (12) 
5 
5 
96 
96 
28 (28) 

1 


44 (68) 
6a 
6a 
96 
96 
36 (36) 

3 


36 (60) 

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
