MS 13. MAGIC SQUARES

DOUBLY MAGIC SQUARES:
Now I come to doubly magic squares: In these squares not only the basic square has the equisum property, but, if the numbers in each cell are replaced by their squares, the resultant square maintains the equisum property. Of course, in the new squares, the numbers will not be in sequence. I reproduce the 8x8 and 9x9 squares with these properties:
05 31 35 60 57 34 08 30
19 09 53 46 47 56 18 12
16 22 42 39 52 61 27 01
63 37 25 24 03 14 44 50
26 04 64 49 38 43 13 23
41 51 15 02 21 28 62 40
54 48 20 11 10 17 55 45
36 58 06 29 32 07 33 59

26 65 32 63 48 15 43 01 76
61 46 13 44 02 77 27 66 33
45 03 78 25 64 31 62 47 14
29 33 71 12 60 54 73 40 07
10 58 52 74 41 08 30 24 72
75 42 09 28 22 70 11 59 53
68 35 20 51 18 57 04 79 37
49 16 55 05 80 38 69 36 21
06 81 39 67 34 19 50 17 56
Here too from this one square a number of new squares can be generated. For example for the 8x8 square if we simultaneously interchange rows 1 & 2, and 7 & 8, and thereafter columns 1 & 2 and 7 & 8, we get the square:
09 19 53 46 47 56 12 18
31 05 35 60 57 34 30 08
22 16 42 39 52 61 01 27
37 63 25 24 03 14 50 44
04 26 64 49 38 43 23 13
51 41 15 02 21 28 40 62
58 36 06 29 32 07 59 33
48 54 20 11 10 17 45 55
It will be seen that combination of numbers in all rows, columns and diagonals is the same, so the resulting square will be doubly magic square. There are more such changes possible. In the case of 9x9 square besides such changes there are additional changes possible. For example 23+58+42=40+24+59, and their squares are also equal being 529+3364+1764=5657=1600+576+3481. Similarly 44+2+77=5+80+38, and their squares are also equal being 7869.

TREBLY MAGIC SQUARES:

It is possible to have magic squares where on replacement of numbers in each cell by the cube of the numbers the resultant square retains the equisum property, but, so far as it is known, we can have it only for the 128_128 square.

OTHER TYPES OF SQUARES;

I had stated that the Magic Squares must have, as one of the conditions, consecutive numbers. I now consider some other types of Squares which have the equisum property.

PRODUCT SQUARES:
We can, certainly, have squares where the sum of the products of the numbers in each row, column or diagonal is the same constant number. All we have to do is to write, instead of the numbers m,m+1,m+2,m+3...........: m,m^2,m^3............

SQUARES WITH ONLY ODD OR EVEN NUMBERS:
We can have squares which have only odd or even numbers in all the cells. Foe odd numbers square, all we have to do, is to increase all even numbers by n^2-1, and for all even number square merely increase all odd numbers by n^2+1, or simply double all the numbers in the cells.