Thursday, December 9, 2010

Another kind of Magic Square

We all know this type of Magic Square ....

8 1 6
3 5 7
4 9 2


where the numbers are arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number

8 + 1 + 6 = 8 + 3 + 4 = 8 + 5 + 2 = 15

For normal magic squares of order n = 3, 4, 5, ..., the magic constants are: n(n^2+1)/2



Now, let's consider a magic square where the numbers 1 to 9 in a 3x3 array so that the numbers surrounding each number add to a multiple of that number.

2 6 5
7 3 1
9 8 4

Notice that
7 + 3 + 6 = 16 (a multiple of 2)
6 + 3 + 1 = 10 (a multiple of 5)
8 + 3 + 1 = 12 (a multiple of 4)
8 + 3 + 7 = 18 (a multiple of 9)

Similarly in the following squares

2 7 9
6 3 8
5 1 4

4 1 5
8 3 6
9 7 2

4 8 9
1 3 7
5 6 2

5 1 4
6 3 8
2 7 9

5 6 2
1 3 7
4 8 9

9 7 2
8 3 6
4 1 5

9 8 4
7 3 1
2 6 5

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