Submitted by Yau King Fung Gabriel 6E (23)

Magic Squares

Submitted by Yau King Fung Gabriel 6E (23)

Rumor has it that long long ago, in about 2200 B.C., when Xia Yu was the emperor, a bizarre thing happened. When he was dealing with the flooding problem, a turtle floated onto the shore in a place called Luo Shui. There was a strange pattern on the turtle's back that makes it unbelievable.

Actually, this pattern is 9 consecutive numbers (1-9) aligned in a 9-box square.

The amazing part is that the sum of the three numbers in a row, in a column or even diagonally is the same, which is 15.

4 / 9 / 2
3 / 5 / 7
8 / 1 / 6

Luo Shu

Fig. A

Researchers further found out the fact that this special square pattern is not limited to this one. So, they named these patterns of consecutive natural numbers, from 1 to n2 , magic squares. Later, researchers discovered that not only consecutive numbers can form magic squares. For example,

(1)  In a 9-box magic square, when a natural number m is added to all of the 9 numbers, a new magic square will be formed.

(2)  In the same 9-box magic square, for number n, when added (n-1)*d where d is any natural number, a new magic square is formed, too.

For each magic square, there is a magic number. For example in the Luo Shu

(Fig.A), the magic number is 15. For example 1, the magic number is 33. For example 2, the magic number is 39. The way to find out the magic number of a number is by the formula:

Magic no. = (a+z)*n/2

where a is the smallest number

z is the largest number

n is the number of rows (columns)

Let's talk about the way to create a magic square by ourselves. For magic squares with odd n (number of rows), we just need to place the numbers in order and then putting them in the right position.

In this way, a 3*3 magic square can be produced. For a 5*5 one, the principle is the same.

Actually, this way of making (odd n*odd n) magic squares is suitable for all odd numbers n. But thing are more difficult for even numbers. For different even numbers, the way of making one is not the same. Here, I only demonstrate the making of 4*4 magic square. For this magic square, the magic number is 34. First, we have the 16 numbers arranged orderly. Then, along the blue and red lines, we have the numbers arranged backwards.

This is the procedure of making a 4*4 magic square.

Besides the ordinary magic squares, there is a kind of special magic square called the diabolic squares. What is special for diabolic squares is that no matter how you cut open the square (vertically or horizontally), and then putting the pieces together, the sum vertically, horizontally and diagonally is still the same.

Here, I give one example of a 4*4 diabolic square.

6 / 13 / 10 / 1
8 / 3 / 4 / 15
5 / 14 / 9 / 2
11 / 0 / 7 / 12
6 / 13 / 10 / 1
8 / 3 / 4 / 15
5 / 14 / 9 / 2
11 / 0 / 7 / 12

The magic number for this diabolic square is 30. After moving the first row to the new position, the magic number is the same, and the square is still a magic square.

Finally, I want to introduce a game about magic squares, which is called the Magic Matrix. The game rule is to choose one number at a time, and cross out numbers in the same row and column of this number. Then, the opponent chooses another number and does the same thing. After several rounds, for example in this 4*4 magic square, 4 numbers will be picked out. The sum of these 4 numbers will always be the same, equal to the maic number. No matter which 4 numbers do the players pick, the sum will still be the magic number 36.

For the first one, 8+8+11+9=36

For the second one, 13+13+7+3=36

For the third one, 14+7+11+4=36

But how to produce a magic square of such kind? In fact, it is just a table of sums. Now, I will show how to do a 4*4 magic matrix. First, I will let the 8 numbers be a,b,c,d,e,f,g,h. No matter which 4 numbers are picked, the sum will always be a+b+c+d+e+f+g+h.

+ / e / f / g / h
a / a+e / a+f / a+g / a+h
b / b+e / b+f / b+g / b+h
c / c+e / c+f / c+g / c+h
d / d+e / d+f / d+g / d+h

For 4*4 magic squares, for a certain 16 numbers, there are 880 ways of arranging them. For 5*5 magic squares, there is more than a million ways. Magic squares are really magical!