The solution for the

Demonstration for the Magic Square 6x6.
(n=6, sum=111)

1. These sizes are not divided by 4. Tow lines remain. So, divide the square into areas shown below. Consider the boundary block as the boundary line.
   

2. Applying the solution of the multiple of 4 to the areas marked A to I, fill in the boxes.

   The sum of the empty boxes of each row or column has to be the number around the square.

   So, 4 blocks(up, down, left, right) have a key to success.

   Let's match sums by switching some numbers in 4 blocks.

3. Fill in the mumbers into corresponding boxes in the 4 'blocks'.

   However, be aware of 2 things:

   >> The numbers in the blocks are temporary as they will be switched around later.
   >> The numbers in the areas marked 'X' are permanent.
   Because they are on the diagonal line, not to be touched.

4. Take the left and right blocks.
   1. Place the right block numbers in reverse order.
      (Think about doing left block numbers..)
      
   2. Switch the numbers in the middle rows between the left and right blocks.
      
   3. Take the number directly below the center line and re-switch them.

5. If you are doing step 4 for up and down blocks, you can't find a pair of numbers to get rid of the difference in step c. Therefore, you must work with some numbers in other areas as below.

   >> Choose one row from areas 'B' or 'H'. And place those numbers in reverse order. (cf. 34 <= => 33)

   >> Choose one row from areas 'D' and take the corresponding row in area 'F'. And place those numbers in reverse order. (cf. 24 <= => 19)

   The sum of each column is different(42-38-36-32). But, by doing step 5, you can find a pair of numbers to get rid of the difference in step c. You can refix the sum of columns to be equal by doing step 6-a. Because the order of sum becomes 42-38-36-32.

6. Take the up and down blocks.
   1. Place the numbers of both blocks in reverse order.
   2. Switch the numbers in the middle columns between the up and down blocks.
   3. Switch the first numbers between the two blocks. (Think about the last numbers..)

   Abracadabra! The Magic Square!!

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Check the Magic Square 10x10 shown below.
(n=10, sum=505)

 1  2  3 94 95 96 97 98  9 10
11 12 13 87 86 85 84 88 19 20
80 29 23 77 76 75 74 28 22 21
70 69 68 34 35 36 37 33 62 61
60 59 58 44 45 46 47 43 52 51
41 42 53 54 55 56 57 48 49 50
40 39 38 64 65 66 67 63 32 31
30 79 73 27 26 25 24 78 72 71
81 82 83 17 16 15 14 18 89 90
 91 92 93  7  6  5  4  8 99 100

Magic Square World

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

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Shin, Kwon Young - brainstm@chollian.net