Numbers: Their Tales, Types, and Treasures.

Chapter 7: Placement of Numbers



A different scheme is used to construct magic squares of singly-even order (i.e., where the number of rows and columns is even but not a multiple of 4). Any singly-even order (say, of order n) magic square may be separated into quadrants (figure 7.14). For convenience, we will label these quadrants as A, B, C, and D.


Figure 7.14: Quadrants of a magic square of singly-even order.

The order of the square should be n, a singly-even number, hence the order of each of the quadrants must be odd. We denote the order of the quadrant by k = 2m + 1 (which is always an odd number for m = 1, 2, 3…). As there is no magic square of order 2, the smallest singly-even magic square will have order 6—in which case m = 1 and k = 3. We have

n = 2k = 2 (2m + 1) = 6, 10, 14, 18…(for m = 1, 2, 3, 4…)

Each of the four quadrants contains k2 different numbers. We start by creating a magic square of odd-order k according to the method described earlier. For n = 6 and k = 3, the starting point will thus be one of the variants of the Lo Shu square. We will choose the first of the magic squares shown in figure 7.12.

We begin by entering this magic square into quadrant A. The magic squares in quadrants B, C, and D will be obtained as shown for the case n = 6 in figure 7.15.


Figure 7.15: First step of the construction of a singly-even magic square of order 6.

Here, square B is obtained by adding k2 = 9 to all numbers of square A. Square C is obtained by adding k2 to all numbers of square B, and square D is obtained by adding k2 to all numbers of square C.

Notice that adding a fixed number to all numbers of a magic square does not change the magic property: The sum of rows, columns, and diagonals would still remain the same. Thus the squares B, C, and D are also magic squares, only they do not use the numbers from 1 to k2. For example, square B uses instead the numbers from k2 + 1 to 2k2 (for n = 6, the numbers 10 to 18). The square of order n obtained in this way is shown in figure 7.16. Although it has magic squares in its quadrants, it is not yet a magic square itself.


Figure 7.16: Second step of the construction of a singly-even-order magic square of order 6.

Continuing along with our construction of the singly-even-order magic square, we have to make some adjustments to the square we have developed to this point. Recall that the integer m determines the order through the formula n = 2 (2m + 1).

In general, the adjustments will be the following: We first take the numbers in the first m positions in each row of quadrant A, except the middle row, where we will skip the first position and then take the next m positions. Then we will exchange the numbers in these positions with the correspondingly placed numbers in square D below. We then take the last m – 1 cells in each row of square C and exchange them with the numbers in the corresponding cells of square B.

For n = 6 and m = 1, the positions in squares A and D that will be changed during that procedure are shaded in figure 7.16. Since, in this case, m – 1 = 0, the squares B and C on the right side remain unaltered. The resulting square is shown in figure 7.17. You may verify that it is indeed a magic square.


Figure 7.17: The singly-even-order magic square obtained from figure 7.16.

We illustrate this procedure once again with the next-larger singly-even magic square, which is of order n = 10, and in this case m = 2.

1.    Starting with a magic square of order 5, we take the one created by the method explained previously (figure 7.13).

2.    We fill the four quadrants of the n × n square. We create square B by adding 25 to all numbers of square A, and then continue as we did earlier. The result is shown as the first square in figure 7.18.

3.    Take the first two positions of each row of quadrant A, except the middle row, where you skip the first cell and then take the next two positions. Exchange the numbers in these cells with the numbers in the corresponding cells of square D. Figure 7.18 has the corresponding positions shaded.

4.    To complete the magic square, we take last m – 1 positions (here the last positions, since m – 1 = 1) in each row of the squares C and B and interchange them. This gives us the magic square shown as the second square in figure 7.18.


Figure 7.18: Construction of a higher-order singly-even magic square.

We now have a procedure for constructing each of the three types of magic squares: the odd-order magic square and both the singly-even and the doubly-even magic squares.

We end this discussion about magic squares with a curiosity, just for entertainment. You can verify that the first square in figure 7.19 is a magic square. The sum of its rows, columns, and diagonals is 45.


Figure 7.19: An alphamagic square.

However, this square has an additional property that makes it a so-called alphamagic square. Replace the numbers by their written words. The number of letters in each word generates a new magic square—the third square in figure 7.19. You can convince yourself of its magic property either by computing all sums of the rows, columns, and diagonals, or by noticing that it also can be obtained from the Lo Shu square by adding two to all its numbers. (Remember, adding a constant number to all numbers of a magic square generates a new magic square.)