MS10. MAGIC SQUARES

SUMMING UP:

We have seen that a 3x3 square only one and 4x4 square 880 solutions and can be constructed by the auxiliary method and also by other methods. A 5x5 square can be constructed either by the Hindu Rule which gives 720 squares from one set, or by the Border Square method with the 3x3 square as the base, or from two auxiliary squares which at once gives 3,600 squares from one set. Further 5x5 squares can be constructed by either constructing 3x3 squares with total of 39 with 13 in the central cell, or by first forming a 3x3 square with any number in the central cell and thereafter filling up the remaining cells with numbers of our choice, except that the total for any one row or column or diagonal should not be more than62. Some illustrations are given below, first 2 with 13 in the central cell:
22 18 10 09 06
07 12 03 24 19
11 25 13 01 15
05 02 23 14 21
20 08 16 17 04

20 01 23 09 12
22 08 21 10 04
02 15 13 11 24
07 16 05 18 19
14 25 03 17 06
Now with 15 in the central cell and different totals for rows and columns for the 3x3 square:
01 10 19 23 12
09 18 22 11 05
17 21 15 04 08
25 14 03 07 16
13 02 06 20 24
It will be seen that the numbers in the border square do not follow any pattern. But we can form additional squares by subtracting all numbers from 26._As is the case with 4x4 square, for 5x5 square too, we can frame rules which will enable us to generate squares from the one already constructed. Let the square be represented by:
a b c d e
f g h i j
k l m n o
p q r s t
u v w x y
We can now generate new squares by interchanging any two columns or rows provided this does not affect the diagonal totals, some examples are given below:
a) If g+s = i+q, we can interchange 2nd and 4th columns
b) If h+n = m+i and m+s=r+n, 3rd and 4th columns.
c) If g+m=h+l and h+n=i+m, 2nd and 3rd rows.
d) If g+m=h+l and l+r=m+q, 2nd and 3rd columns.
e) If l+r=m+q and m+s=r+n, 3rd and 4th rowsw.
f) Also if b+c=v+w, they can be interchanged in the 1st and 5th rows, etc.
g) As stated earlier new squares can be generated merely by subtracting all numbers from 26. These rules, incidentally, have applicability in general for all squares of any order.