Monday, May 2, 2011

3x3 Magic Square

Last time we looked at magic squares in general and found out that the most simple magic square is of order 3x3.

There is beautiful legend about the discovery of this popular magic square.

As the story goes, in ancient China there was a huge deluge: the people offered sacrifices to the god of one of the flooding rivers, the Lo river (洛水), to try to calm his anger. A magical turtle emerged from the water with the curious and decidedly unnatural (for a turtle shell) Lo Shu pattern on its shell, which contained the numbers in an aray of a magic square. As the legend says, these numbers showed the people how often and how many sacrificies should be given to calm the Lo river.

The Lo Shu square containing the numbers on the turtle's back:

So what are these magic numbers?

It appears that all magic squares obey some rules. For instance all normal magic squares of un-even order (of  n x n order, where n = 3,5,7...) have a special number at the centre called the magic constant.

So for a 3x3 magic square you have 1+2+3+4+5+6+7+8+9=45. In a magic square you have to add 3 numbers again and again. Therefore the average sum of three numbers is 45:3=15. The number 15 is called the magic number of the 3x3 square.
You can also achieve 15, if you add the middle number 5 three times.  You can reduce 15 in a sum of three sums eight times: 15= 1+5+9;  15= 2+4+9;  15= 2+6+7;  15= 3+5+7

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

The odd numbers 1,3,7, and 9 occur twice in the reductions, the even numbers 2,4,6,8 three times and the number 5 four times. Therefore you have to place number 5 in the middle of the magic 3x3 square.

The remaining odd numbers have to be in the middles of a side and the even numbers at the corners. 

*The formula to find other magic constants is:

M= [n(n^2 +1)]/2

So by using the information above we can see that there exists only 1 magic 3x3 square (of numbers 1-9). Of course you can rotate the magic square in total of 8 ways, which gives all the possible distributions of the numbers in a 3x3 magic square.

Thanks for reading!


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