MS 12. MAGIC SQUARES

TYPES OF SQUARES - BREAK-UP:
The split of the 880 squares by type as defined by us is:
Regular Squares - 528.
Type A - 120, 3 numbers <9 in one diagonal and 3 numbers >8 in the other.
Type B - 64, 2 numbers <9 and 2 numbers>8 in each row, column and diagonal, but not a Regular Square.
Type C - 120, all odd numbers in one diagonal and all even numbers in the other.
Type AB - 48, all odd numbers in one diagonal and all even numbers in the other, but otherwise Type A.

MAGIC SQUARES FROM MAGIC SQUARES:
We have seen earlier how we can construct a large number of squares of order 5 and above. We now discuss another special and interesting method for construction of 'p' order squares, where p = mxn, with both m and n being eaul to or greater yhan 3. The method requires that we replace the integers by the squares of of the other order. We illustrate this method for the 12x12 square, 12 being equal to 3x4.
The 12x12 square will have 144 cells, numbers to be used will be from 1 to 144, and the magic sum will be 145x6=870. Imagine these 144 cells being divided into 9 compartments having 16 adjacent cells in 4 rows and 4 columns. If we allot to the compartment numbers from 1 to 9 as if we are constructing a 3x3 square, all we need to do is to replace these numbers 1 to 9 by 4x4 squares. This is what we have to do:
1 is to be replaced by 4x4 square with numbers from 1 to 16 and total 34.
2 is to " " " " " 17 to 32 and total 64+34=98.
3 is to " " " " " 33 to 48 and total 128+34=162.
......................................... 9 is from 129 to 144, total 512+34= 546.
Now in the 3x3 square one set of diagonal is 4,5&6, so we get the for the mail diagonal a total of 192+34+256+34+320+34= 870, the required total for a 12x12 square. All numbers from 1 to 144 have been used once only etc. While there is only one solution for the 3x3 square (the square however can be rotated) there are 880 choices for each of the 9 squares we have to use, and since all of them can be rotated we have in all 880x4^9 squares that can be obtained, each different.
Alternately we divide the 144 cells into 16 compartments numbering from 1 to 16, forming the 4x4 square, all we have to do now is replace 1 by a 3x3 square with numbers from 1 to 9, 2 by a 3x3 square with numbers from 10 to18, and so on...
As we can use 880 squares for the 4x4 square we have 880x4^16 squares. Of course, for odd number squares the Hindu Rule is always there and for singly even order squares the border square method.

MORE ABOUT 8X8 SQUARE:
Before we proceed further note that there are exceptions. The 8x8 square can be constructed with the help of 4x4 squares. This is possible because the total required is 260, and so if we have 4 compartments of 4x4 squares each having a total of 130, we get the 8x8 squarte. The only restriction here would be that we cannot use all the 880 squares but 712 squares. (We cannot use squares of Type A and Type AB for obvious reasons).

MORE ABOUT 12X12 SQUARES:
In the same manner we can also construct 12x12 squares from 6x6 squares, provided we construct the 6x6 squares by De La Hire's method. The 4 squares required are to be formed by adding:
0 to numbers from 1 to 18 and 108 to numbers from 19 to 36.
18 to " 1 to 18 and 90 to " from 19 to 36.
36 to " 1 to 18 and 72 to " from 19 to 36.
54 to all numbers.
Note: For the last square all the 880 squares can be used as the same number 54 is being added to all numbers.
We can similarly form other squares also with due precaution of selecting the right squares.