## Numbers: Their Tales, Types, and Treasures.

## Chapter 7: Placement of Numbers

### 7.4.CONSTRUCTION OF A MAGIC SQUARE OF ORDER 3

A systematic construction of all possible 3 × 3 magic squares would begin by considering the matrix of letters representing the numbers 1 to 9 shown in __figure 7.9__. Here the sums of the rows, columns, and diagonals are denoted by *r** _{j}*,

*c*

*, and*

_{j}*d*

*, respectively. In a magic square of order 3, all these number sums would be equal to the magic number 15.*

_{j}**Figure 7.9: A map of a general magic square**

In a magic square, we would thus have

*r*_{2} + *c*_{2} + *d*_{1} + *d*_{2} = 15 + 15 + 15 + 15 = 60.

However, this sum can also be written as

*r*_{2} + *c*_{2} + *d*_{1} + *d*_{2} = (*d* + *e* + *f*) + (*b* + *e* + *h*) + (*a* + *e* + *i*) + (*c* + *e* + *g*) =

3 *e* + (*a* + *b* + *c* + *d* + *e* + *f* + *g* + *h* + *i*) = 3*e* + 45

Therefore, 3*e* + 45 = 60, and *e* = 5. Thus it is established that the center position of a magic square of order 3 must be occupied by the number 5.

Recall that two numbers of an *n*th order magic square are said to be complementary if their sum is *n*^{2} + 1. In a 3 × 3 magic square, two numbers are complementary if their sum is 9 + 1 = 10. We can now see that numbers on opposite sides of 5 are complementary. For example, *a* + *i* = *d*_{1}– *e* = 15 – 5 = 10, and, therefore, *a* and *i* are complementary. But so are the pairs *g* and *c*, *b* and *h*, and *d* and *f*.

Let us now try to put 1 in a corner, as shown in __figure 7.10__. Here *a* = l, and therefore *i* must be 9, so that the diagonal adds up to 15. Next we notice that 2, 3, and 4 cannot be in the same row (or column) as 1, since there is no natural number less than 9 that would be large enough to occupy the third position of such a row (or column). This would leave only the two shaded positions in __figure 7.10__ to accommodate these three numbers (2, 3, and 4). Since this cannot be the case, our first attempt was a failure: the numbers 1 and 9 may occupy only the middle positions of a row (or column).

**Figure 7.10: A non-magic-square construction—false start.**

Therefore, we have to start with one of the four possible, positions remaining for 1, for example, as we show in the first square of __figure 7.11__. We note that the number 3 cannot be in the same row (or column) as 9, for the third number in such a row (or column) would again have to be 3 to obtain the required sum of 15. This is not possible, because a number can be used only once in the magic square. Additionally, we have seen above that 3 cannot be in the same row (or column) as 1. This leaves only the two shaded positions in __figure 7.11__ for the number 3. The number opposite 3 is always 7, because then 3 + 5 + 7 = 15.

**Figure 7.11: The development of one of several possible magic squares.**

We continue with the second square in __figure 7.11__, showing one of two possibilities for the placement of 3 and 7 (the other possibility has 3 and 7 exchanged). It is now easy to fill in the remaining numbers. There is only one such possibility, shown in the third square of __figure 7.11__.

How many different squares are there? We could start by putting the number 1 in any of the four positions in the middle of a side. We then have two possibilities for placing 3. After that, the construction is unique. This produces the eight magic squares shown in __figure 7.12__.

**Figure 7.12: There are precisely eight magic squares of order 3.**