MS 14. MAGIC SQUARES

COMBINATION OF 4X4 AND 3X3 SQUARE:
Here is another illustration of what we can do with these squares. Study the square below which has 4 numbers and 3 numbers in alternate rows.
30 00 44 00 43 00 33
00 49 00 54 00 47 00
41 00 35 00 36 00 38
00 48 00 50 00 52 00
37 00 39 00 40 00 34
00 53 00 46 00 51 00
42 00 32 00 31 00 45
The 4X4 square totals 150 and has numbers from 30 to 45. The 3X3 square totals 150 too and has numbers from 46 to 54. The main diagonals total 300 as expected.

UPSIDE DOWN SQUARE:
Another interesting square which totals the same read upside down and has_multiple solutions is shown below:
96 11 89 68
88 69 91 16
61 86 18 99
19 98 66 81
To get multiple solution all we have to do is for the alphabets A, B, C & D and a, b, c & d take the numbers 1,6,8,9. (Please note here Aa is not to be taken as A+a).

STAR-SHAPED SQUARE:
Some elegant constructions on star shaped figures (pentagons, hexagons, etc.,) can also be developed. An instance of octagon is illustrated below:
- - - - - - - 01 - - - - - - - -
- - - 02 - 11 - - 16 - - 05 - - -
- - - - 12 - - - - - 09 - - - - -
- 10 - - - - - - - - - - - - - 08
- - - - 06 - - - - - 07 - - - - -
- - - 14 - 03 - - 04 - - 13 - - -
- - - - - - - 15 - - - - - - - -
There are magic circles, rectangles, crosses and diamonds, which can be developed for fun!

CHESS BOARD MOVES:
Attempts have also been made to develop a 8_8 square based on the KNIGHT'S MOVES on the chess board. So far nobody has been successful. The square, as worked out, does not give the magic sum of 260 for the diagonals. See the square below:
47 10 23 64 49 02 59 06
22 63 48 09 60 05 50 03
11 46 61 24 01 52 07 58
62 21 12 45 08 57 04 51
19 36 25 40 13 44 53 30
26 39 20 33 56 29 14 43
35 18 37 28 41 16 31 54
38 27 34 17 32 55 42 15

SQUARES FOR ANY GIVEN NUMBER:
We can easily construct magic squares for a given calendar year or for any other number, but not always. For example for the year 1892 or 2002 we can construct a 11_11 square as both these numbers are divisible by 11. All we have to do is to start with 112 for getting 1892 and 122 for getting 2002._We can also construct 4_4 squares for odd magic sum. To get 35 add 1 to 13,14,15,&16. To get 36 add 2 etc. Of course for this it is necessary that we select a square in which 13,14,15&16 are so distributed that they appear in all rows, all columns and the diagonals.

POLYHEDRONS:
Polyhedrons are defined as solid objects with 6 or more surfaces. Hexahedron has 6 surfaces and is popularly called a cube. We can decide to have same order square on all sides or, if we so like, have a different order squares on each side. For example for a Octahedron, (which has eight sides), if we decide to have on all sides 4x4 squares with identical total, we will be required to use the numbers from 1 to 128 and the equisum will have to be 258. All we are required to do first is to select a 4x4 square which has 2 numbers less than 9 and two numbers greater than 8. Next we have to get the 8 squares by adding 0 to all numbers from 1 to 8 and 112 to all numbers from 9 to 16, 8 to all numbers from 1 to 8 and 104 to all numbers from 9 to 16, 16 to all numbers from 1 to 8 and 96 to all numbers from 9 to 16, 24 to all numbers from 1 to 8 and 88 to all numbers from 9 to 16, 32 to all numbers from 1 to 8 and 80 to all numbers from 9 to 16, 40 to all numbers from 1 to 8 and 72 to all numbers from 9 to 16, 48 to all numbers from 1 to 8 and 64 to all numbers from 9 to 16, 56 to all numbers from 1 to 8 and 56 to all numbers from 9 to 16.
If we decide to have squares of, say, order 8 on all sides, we require numbers from 1 to 512 with equisum of 2052, follow a similar procedure increasing numbers from 1 to 32 by 0, 32, 64 ...... and increasing numbers from 33 to 64 by 448,416, 384.... If, however, we decide on a 4x4 order square on one side, 5x5 order square on second side, 6x6 order square on the third side..... there is no problem except that totals on each side will depend on the numbers selected for each square. If we want a uniform total for all sides that will be possible but we may have to deal with very large numbers.

HYPER - MAGIC SQUARES:
I have stated earlier that we can impose varying conditions for these squares. I will now discuss some of these, they will be found only in some of the squares.

DIABOLIC SQUARES:
A Magic Square is called Diabolic, Perfect or Pandiagonal. It is diabolic if it has the same constant sum for its broken diagonals as it has for its rows, columns and diagonals. Examine the square:
01 08 11 14
12 13 02 07
06 03 16 09
15 10 05 04
The broken diagonals are; 1 & 10, 16, 09; 4 & 11, 13, 06; 12, 08 & 05, 09; 06, 10 & 11, 07; 15 & 08, 02, 09; 14 & 12, 03, 05. The square from ancient India and the Jaina square also have this property. I also reproduce below a 5x5 square with this property.
01 08 15 17 24
20 22 04 06 13
09 11 18 25 02
23 05 07 14 16
12 19 21 03 10
Obviously there can't be a diabolic square of order 3. Also there are no diabolic singly even squares like 6, 10, etc. An 8x8 square constructed from diabolic square of order 4 will also be diabolic. One property of diabolic squares, worth mentioning, is that if a pandiagonal square be cut into 2 pieces along a line between any two rows or columns, the pieces be interchanged, the new square, so formed, will also be a magic square and retain the property of being diabolic. This gives us another method for generating squares from an existing square. Reproduced below is a 5_5 square from the one shown above:
08 15 17 24 01_22 04 06 13 20_11 18 25 02 09_05 07 14 16 23_19 21 03 10 12

ASSOCIATIVE SQUARES:
An Associative or Symmetrical square is one that has pair of numbers symmetrically opposite the center and add up to n^2+1. A 4x4 square may be associative or diabolic, but not both, but an odd square order can be both (in that case the center number has necessarily to be in the center). Both a 4x4 square and 5x5 square with this property are shown below:
05 04 16 09
01 15 24 08
11 14 02 07
08 01 13 12

17 10 15 03 06
23 07 16 05 14
20 04 13 22 06
12 21 10 19 03
09 18 02 11 25

REGULAR SQUARES:
Just as we have regular squares of order 4, we can also have regular squares of order 8, 16, 32, 64, etc. The procedure is the same as for a 4x4 square.