Fellows Transformations

Introduction

In August 1999 I received the first of several communications from Ralph Fellows detailing work he was doing with transformations. with his permission I present some of his methods on this page.

The ideas are his. However, I have reworded the descriptions and have adapted examples to order-4 magic squares to conform with the other pages of this section.
In the examples, I use leading 0's for the 1 digit decimal integers to provide easier formatting.

A period-4 loop

If the number is 1, 5, 9 or 13 then add 3 to the number, else subtract 1 from the number. Example

#93    IV     #660    V      #467   IV    #331    V     #93    IV
01 07 14 12   04 06 13 11   03 05 16 10   02 08 15 09   01 07 14 12
16 10 03 05   15 09 02 08   14 12 01 07   13 11 04 06   16 10 03 05
11 13 08 02   10 16 07 01   09 15 06 04   12 14 05 03   11 13 08 02
06 04 09 15   05 03 12 14   08 02 11 13   07 01 10 16   06 04 11 15

There is no group in which this works for all magic squares. Mr. Fellows claims it works for 112 of the 880 order-4 magic squares.

For order-5, add 4 to 1, 6, 11, 15, 21 and subtract 1 from all the other numbers gives a loop of five.
For order-6, add 5 to 1, 7, 13, 19, 25, 31 and subtract 1 from the other numbers for a loop of six. Etc.

1. Change magic square to a magic square using numbers from 0 to 15 by subtracting 1 from each number.
2. Change to base four numbers.
3. Swap the two digits of each number.
4. Change back to base ten.
5. Change back to numbers 1 to 16.

#238    X     A. 0 to 15    B. base 4     C. swap dig.  D. base 10    E. #794  X
02 04 13 15   01 03 12 14   01 03 30 32   10 30 03 23   04 12 03 11   05 13 04 12
16 14 01 03   15 13 00 02   33 31 00 02   33 13 00 20   15 07 00 08   16 08 01 09
05 07 12 10   04 06 11 09   10 12 23 21   01 21 32 12   01 09 14 06   02 10 15 07
11 09 08 06   10 08 07 05   22 20 13 11   22 02 31 11   10 08 07 05   11 09 08 06

Shortcut: Instead of working out these 5 steps in turn, the numbers of the original magic square may be converted directly to the following decimal numbers.

original number: 1. 2. 3... 4. 5. 6.. 7.. 8. 9. 10 11 12 13 14 15 16
new number: .....1. 5. 9. 13. 2. 6 10 14. 3.. 7. 11 15.. 4.. 8 12 16

This transformation works for all groups I to VI-P. It also works for some magic squares of groups VI-S to X, but no magic squares of groups XI or XII.
Sometimes for groups I, II and III the result is a 180 degree rotation of the original i.e. a self-similar magic square. When the result is a new magic square, it is always a member of the same group as the original.

This table shows the index number of a magic square in each group that is magic, not magic or a reflection.

Note the similarity of this table to the one for the next two transformations. However, all 3 transformations generate different magic squares from the same original.
There are 18 groups of four numbers that when converted as per step E above, do not sum to 34. Nine of these sum to 19 and 9 of these sets sum to 49. This also applies to the next 2 transformations, complement LSD and complement MSD of the base 4 number.
There is only one condition to search for when looking for a possible rotation. One of the two main diagonals must contain the numbers 1, 6, 11 and 16 (in any order). The reason for this is because the base 4 representation of each of these numbers when they are reduced by 1 is 00, 11 22 and 33. This results in the line containing the same four numbers when the digits are reversed.

The digit swap method works for other magic square orders by first converting the magic square to the base for that order.

Complementing LSD or MSD

Complement only the least significant digit of the base 4 numbers.

#126          0 to 15       base 4        comple. LSD   back to dec.  + 1 = #632 
01 08 15 10   00 07 14 09   00 13 32 21   03 10 31 22   03 04 13 10   04 05 14 11
14 11 04 05   13 10 03 04   31 22 03 10   32 21 00 13   14 09 00 07   15 10 01 08
12 13 06 03   11 12 05 02   23 30 11 02   20 33 12 01   08 15 06 01   09 16 07 02
07 02 09 16   06 01 08 15   12 01 20 33   11 02 23 30   05 02 11 12   06 03 12 13

This works for Types I to X.
Types XI and XII have correct rows and columns, but both diagonals are incorrect.

Complement only the most significant digit of the base 4 numbers.

#126          0 to 15       base 4        comple. MSD   back to dec.  + 1 = #632
01 08 15 10   00 07 14 09   00 13 32 21   30 23 02 11   12 11 02 05   13 12 03 06
14 11 04 05   13 10 03 04   31 22 03 10   01 12 33 20   01 06 15 08   02 07 16 09
12 13 06 03   11 12 05 02   23 30 11 02   13 00 21 32   07 00 09 14   08 01 10 15
07 02 09 16   06 01 08 15   12 01 20 33   22 31 10 03   10 13 04 03   11 14 05 04

Note that this example results in a rotated copy of that obtained from the previous example. However, sometimes the result is a different magic square then that obtained by complementing the LSD. Starting with #303, type VII, #486 was obtained by complementing the LSD and #510 by complementing the MSD.

For a shortcut when working with these two transformations, substituting the new decimal integer directly for the old one bypasses the four intermediate steps.

Original number: 1  2  3  4  5  6  7  8  9  10 11 12 13  14 15 16
New number
LSD Transform:   4  3  2  1  8  7  6  5  12 11 10  9 16  15 14 13
MSD Transform:   13 14 15 16 9 10 11 12  5  6  7   8  1   2  3  4

Complementing both base 4 digits is the same as complementing the original decimal number.

All 3 transformations generate different magic squares from the same original.
In all cases, the new magic square is part of the same group as the original magic square.
This method also works for other magic square orders by first converting the magic square to the base for that order.

Increment base 4 digits

This transformation works by cycling through the base 4 digits, incrementing by 1 (with no carry) then changing back to decimal. This first section illustrates the six steps required for each transition. I use leading 0’s in the decimal magic squares simply for formatting.

A             B             C             D             E             F
104    I      -->0-15       Base 4        MSD + 1       Dec 0 to 15   508    IV
01 08 10 15   00 07 09 14   00 13 21 32   10 23 31 02   04 11 13 02   05 12 14 03
14 11 05 04   13 10 14 03   31 22 10 03   01 32 20 13   01 14 08 07   02 15 09 08
07 02 16 09   06 01 15 09   12 01 33 20   22 11 03 30   10 05 03 12   11 06 04 13
12 13 03 06   11 12 02 05   23 30 02 11   33 00 12 21   15 00 06 09   16 01 07 10

I will start a loop of 16 transformations by starting with step C then step F (above).
Loop starts. by incrementing square C (the representation of magic square 104), show the resulting magic square in decimal using integers 1 to 16, then next increment of the base 4 square, etc.

MSD+0,LSD+1   250   IV      MSD+0,LSD+1   473    I      MSD+0,LSD+1   669   IV
01 10 22 33   02 05 11 16   02 11 23 00   03 06 12 13   03 12 20 31   04 07 09 14
32 23 11 00   15 12 06 01   33 20 12 01   16 09 07 02   30 21 13 02   13 10 08 03
13 02 30 21   08 03 13 10   10 03 31 22   05 04 14 11   11 00 32 23   06 01 15 12
20 31 03 12   09 14 04 07   21 32 00 13   10 15 01 08   22 33 01 10   11 16 02 05

MSD+1,LSD+0   308   III     MSD+0,LSD+1   508   IV      MSD+0,LSD+1   632   III
13 22 30 01   08 11 13 02   10 23 31 02   05 12 14 03   11 20 32 03   06 09 15 04
00 31 23 12   01 14 12 07   01 32 20 13   02 15 09 08   02 33 21 10   03 16 10 05
21 10 02 33   10 05 03 16   22 11 03 30   11 06 04 13   23 12 00 31   12 07 01 14
32 03 11 20   15 04 06 09   33 00 12 21   16 01 07 10   30 01 13 22   13 02 08 11

MSD+0,LSD+1   65   IV       MSD+1,LSD+0   281    I      MSD+0,LSD+1   504   IV
12 21 33 00   07 10 16 01   22 31 03 10   11 14 04 05   23 32 00 11   12 15 01 06
03 30 22 11   04 13 11 06   13 00 32 21   08 01 15 10   10 01 33 22   05 02 16 11
20 13 01 32   09 08 02 15   30 23 11 02   13 12 06 03   31 02 12 03   14 09 07 04
31 02 10 23   14 03 05 12   01 12 20 33   02 07 09 16   02 13 21 30   03 08 10 13

MSD+0,LSD+1   623    I      MSD+0,LSD+1   60    IV      MSD+1,LSD+0   478   III
20 33 01 12   09 16 02 07   21 30 02 13   10 13 03 08   31 00 12 23   14 01 07 12
11 02 30 23   06 03 13 12   12 03 31 20   07 04 14 09   22 13 01 30   11 08 02 13
32 21 13 00   15 10 08 01   33 22 10 01   16 11 05 02   03 32 20 11   04 15 09 06
03 10 22 31   04 05 11 14   00 11 23 32   01 06 12 15   10 21 33 02   05 10 16 03

MSD+0,LSD+1   682   IV      MSD+0,LSD+1   126   III     MSD+0,LSD+1   274   IV
32 01 13 20   15 02 08 09   33 02 10 21   16 03 05 10   30 03 11 22   13 04 06 11
23 10 02 31   12 05 03 14   20 11 03 32   09 06 04 15   21 12 00 33   10 07 01 16
00 33 21 12   01 16 10 07   01 30 22 13   02 13 11 08   02 31 23 10   03 14 12 05
11 22 30 03   06 11 13 04   12 23 31 00   07 12 14 01   13 20 32 01   08 09 15 02

The next iteration would be MSD +1, LSD+0, which would give us square C in the example above. This square represents #104, group I, which is the magic square we started with.
Notice that there are 4 group I and 4 group III, but 8 group IV. It would seem that this works out correctly because there are 48 magic squares of groups I and III but 96 magic squares of group IV.
However, when I worked out another loop (starting with # 109), it contained 4 group I, 4 group IV, 4 group V and 4 group VI-P. Also, this method does not work for many magic squares (including group I - 102, 116, and 279).

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Last updated September 09, 2003
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 2000 by Harvey D. Heinz