Oddorder magic squares expressed in J
John C. McInturff
This note illustrates an arrayoriented approach to solving oddordered magic squares. Part 1 illustrates a computersensible solution expressed in J. The resulting solution is then subjected to eight symmetrical transformations and each tested for ‘magic properties’. Part 2 describes the underlying twostep rule for the solution, and illustrates how this rule can be applied to the entire magic square (matrix), and carried out graphically without a computer. Each graphical step is made computersensible and is executed.
Part 1
The following verb, MS
is an arrayoriented solution to
an oddordered Magic Square, expressed in J. This expression is
intended to minimise keystrokes, not maximise the understanding of
the thought behind it. The latter objective is addressed in Part 2.
MS=. 3 : ' (1. N) P : (N=. ,.i.y) (P=. ."1) (<.:y) . >:i.2# y'
Shown below are four examples, for n=. 3 5 7 9
.
The verb, each, is: ea=. &.>
MS ea 3 5 7 9 ++ ¦8 1 6¦17 24 1 8 15¦30 39 48 1 10 19 28¦47 58 69 80 1 12 23 34 45¦ ¦3 5 7¦23 5 7 14 16¦38 47 7 9 18 27 29¦57 68 79 9 11 22 33 44 46¦ ¦4 9 2¦ 4 6 13 20 22¦46 6 8 17 26 35 37¦67 78 8 10 21 32 43 54 56¦ ¦ ¦10 12 19 21 3¦ 5 14 16 25 34 36 45¦77 7 18 20 31 42 53 55 66¦ ¦ ¦11 18 25 2 9¦13 15 24 33 42 44 4¦ 6 17 19 30 41 52 63 65 76¦ ¦ ¦ ¦21 23 32 41 43 3 12¦16 27 29 40 51 62 64 75 5¦ ¦ ¦ ¦22 31 40 49 2 11 20¦26 28 39 50 61 72 74 4 15¦ ¦ ¦ ¦ ¦36 38 49 60 71 73 3 14 25¦ ¦ ¦ ¦ ¦37 48 59 70 81 2 13 24 35¦ ++
The sum of each row, column, right diagonal, and left diagonal, is
required to be equal to the value known as the magic
value produced by the verb val=. [::]*1+*:
.
The magic value for each magic square above is therefore:
val ea 3 5 7 9 ++ ¦15¦65¦175¦369¦ ++
The question is: do the above matrices satisfy the ‘magic requirement’? The answer requires n calculations where n is equal to the magic value plus the two diagonals plus double the number of sides for each matrix. For matrices of order 3, 5, 7, 9 and 41, n would be:
]n=. (1+2++:) ea 3 5 7 9 41 ++ ¦9¦13¦17¦21¦85¦ ++
The objective now is to see if the four matrices meet the above conditions. The sum of each row, column, left diagonal, and right diagonal is given by the following verbs:
row=. +/"1 col=. +/"2 d1=.[: +/ (<0 1) : ] d2=. [: +/ (<0 1) : . v=. [: val #
For brevity, the following verb f
combines the above 5
verbs and illustrates an example of its use for n=. 5
f=. (v;' ';row;col;d1;d2) f (MS 5) ++ ¦65¦ ¦65 65 65 65 65¦65 65 65 65 65¦65¦65¦ ++
It is seen that the order5 matrix took thirteen calculations and
met all conditions. An order41 matrix would require 85 conditions.
The following verb Test
, when applied to the matrix
order, takes all of the above requirements into account and returns
a 1 if all conditions are met; e.g.,
Test=. 3 : 0 n=. val y q=. MS y *./n=(,>(row q)),(col q)),(d1 q),(d2 q) )
This will now be illustrated for all odd matrices from order 3 through 41.
odd=. (1+2*i.21) n=.(3++:) odd test=. Test ea odd >< ea odd,n,:(>test) ++ ¦1¦3¦5 ¦7 ¦9 ¦11¦13¦15¦17¦19¦21¦23¦25¦27¦29¦31¦33¦35¦37¦39¦41¦ +++++++++++++++++++++¦ ¦5¦9¦13¦17¦21¦25¦29¦33¦37¦41¦45¦49¦53¦57¦61¦65¦69¦73¦77¦81¦85¦ +++++++++++++++++++++¦ ¦1¦1¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦1 ¦ ++
The value of a magic square is unaffected by the following eight
transformations. These are the identity transformation
t0
, plus three clockwise rotations and their four
respective reflections. There are, therefore, eight magic squares
associated with the verb MS
.
These eight transformations are expressed by the following eight verbs and are illustrated below:
t0=: ] t1=: t6@t7 t2=: t4@t6 t3=: .@: t4=: .@] t5=: t2@t7 t6=: ."_1@] t7=: :@]
For brevity, these eight transformations are combined into the
single verb, t
, and are more clearly illustrated below
for the rectangular 3×2 matrix, a
.
t=. (t0;t1;t2;t3;t4;t5;t6;t7) ]a=. >:i.3 2 1 2 3 4 5 6
]T=. t a ++ ¦1 2¦5 3 1¦6 5¦2 4 6¦5 6¦6 4 2¦2 1¦1 3 5¦ ¦3 4¦6 4 2¦4 3¦1 3 5¦3 4¦5 3 1¦4 3¦2 4 6¦ ¦5 6¦ ¦2 1¦ ¦1 2¦ ¦6 5¦ ¦ ++
The eight magic squares are therefore:
t (MS 3) ++ ¦8 1 6¦4 3 8¦2 9 4¦6 7 2¦4 9 2¦2 7 6¦6 1 8¦8 3 4¦ ¦3 5 7¦9 5 1¦7 5 3¦1 5 9¦3 5 7¦9 5 1¦7 5 3¦1 5 9¦ ¦4 9 2¦2 7 6¦6 1 8¦8 3 4¦8 1 6¦4 3 8¦2 9 4¦6 7 2¦ ++
Part 2
The third magic square of the eight above is the 650BCE Lo Shu magic square[1], often credited as being the first recorded magic square. (Somehow it got on the back of a turtle!)
The French diplomat Simon de la Loubère 16421749[2] published the following “well known” rule for solving oddordered magic squares.
 Initialise a square grid (matrix) by placing the integer 1 in the center column of the first row.

Place the next number, 2, in the square diagonally up and to the right.
 If filled, move vertically down one square,
 If ‘off the square’, wrap around (odometerwise) to the last row, or first column, respectively.
 Continue with the next number 3 etc. (repeating the above rule if necessary) until the square is filled.
One can start with any number other than 1 and follow the above rule to derive other magic squares belonging to the group of eight mentioned in Part 1.
Although the above rule involves a single number at a time and an iterative process, it can be carried out extremely rapidly by hand. Furthermore, it is the basis for the thought process that takes place when one applies the same principle to the matrix as a whole and follows the same twostep rule.

The first step is to initialise the matrix such that the integer
1 will always appear in the centre column of an outside edge.
This is assured by verb
f2
in the Appendix.  The second step is to move the entire matrix diagonally up and to the right. The term move is a vector instruction that successively shifts each row of the matrix, as illustrated in the Appendix.
Two important comments must be made:

For the arrayoriented language used, it may be more desirable
to ‘translate’ the matrix and move equivalently,
left and up as was done in the verb
MS
. 
Left and up can be thought of as two degrees
of freedom. If one has only one degree of freedom, it can be
equivalently accomplished by an anticlockwise translation and a
left shift as was done in the verb
MS
.
The array approach using the above method can be carried out by hand as well as executed by a computer as described in detail in the Appendix.
References

Lo Shu Square
http://en.wikipedia.org/wiki/Lo_Shu_Square 
Simon de la Loubère
http://en.wikipedia.org/wiki/Simon_de_la_Loub%C3%A8re
Appendix
Graphical  Computer  











