Analysis for a Multiple of 4 Series

- All of the 8 x 8 Magic Square -
.

1. Regular Magic Squares according to formal pattern.

See figures shown below. The left figure is used at 'A multiple of 4 series' page, and the right one is used at other sites or books.
The common laws drawn out from two figures are

So, you design just one quarter as same numbers of grays and whites are in each row and column. And then reflect the pattern to neighbour quarter.
Now, write numbers in there corresponding cells of areas painted gray. And place the remaining numbers in reverse corresponding order from the end.
Next examples help you to understand.


  1. 1 63 3 61 60 6 58 8
    56 10 54 12 13 51 15 49
    17 47 19 45 44 22 42 24
    40 26 38 28 29 35 31 33
    32 34 30 36 37 27 39 25
    41 23 43 21 20 46 18 48
    16 50 14 52 53 11 55 9
    57 7 59 5 4 62 2 64


  2. 1 63 62 4 5 59 58 8
    9 10 54 53 52 51 15 16
    48 18 19 45 44 22 23 41
    40 39 27 28 29 30 34 33
    32 31 35 36 37 38 26 25
    24 42 43 21 20 46 47 17
    49 50 14 13 12 11 55 56
    57 7 6 60 61 3 2 64


  3. 64 2 3 61 60 6 7 57
    9 10 54 53 52 51 15 16
    48 47 19 20 21 22 42 41
    25 39 38 28 29 35 34 32
    33 31 30 36 37 27 26 40
    24 23 43 44 45 46 18 17
    49 50 14 13 12 11 55 56
    8 58 59 5 4 62 63 1

2. Magic Squares modified by switching some positions.

1 63 3 61 60 6 58 8
56 10 54 12 13 51 15 49
17 47 19 45 44 22 42 24
40 26 38 28 29 35 31 33
32 34 30 36 37 27 39 25
41 23 43 21 20 46 18 48
16 50 14 52 53 11 55 9
57 7 59 5 4 62 2 64
This table is the same thing with example-1. ..How about switching the number '56' & '16' ?
2nd row and 7th row go to different sum. But switching the number '12' & '52'(or '13' & '53') makes the sum equal. It means one more Magic Square is created. (By the way, choosing '10' & '50' or '15' & '55' is not good because of diagonal sum.)
Let's switch one more. If you switch '26' & '31' in gray cells and switch '23' & '18' in white cells, you will make another Magic Square. ('23' is on the same column with '26', and also '31' is on the same column with '18'.)
Can you do it?
1 63 3 61 60 6 58 8
16 10 54 52 13 51 15 49
17 47 19 45 44 22 42 24
40 31 38 28 29 35 26 33
32 34 30 36 37 27 39 25
41 18 43 21 20 46 23 48
56 50 14 12 53 11 55 9
57 7 59 5 4 62 2 64

3. More Magic Squares

I research a fundamental principle for simple and formal Magic Squares, and then handle irregular Magic Squares.
But I give up trying to switch for diagonal, or crossing, or twisting, or any positions. My mathematical ability doesn't find out how many 8x8 Magic Squares are. It's yours..

Anyway, to simplify the model is important to understand a fundamental principle like a scientific researching process. So, It is better applying the simple pattern(as shown right) to the solution of the other series.


Magic Square World

|| Solution(n=3,5,..)|| Solution(n=4,8,..) || Solution(n=6,10,..) || Source Program || Samples ||

Shin, Kwon Young - brainstm@chollian.net