The Moscow Puzzles: 359 Mathematical Recreations by Boris A. Kordemsky

Author:Boris A. Kordemsky [Kordemsky, Boris A.]
Language: eng
Format: epub
ISBN: 9780486801308
Publisher: Dover Publications
Published: 0101-01-01T00:00:00+00:00

Form magic squares of the third and (on your own) seventh orders, using the method just described.

Squares of orders that are multiples of 4: Here is one rather easy method:

1. Number the cells consecutively, as shown in the 4-by-4 square (a below) and the 8-by-8 square (c).

2. Divide the square with two vertical and two horizontal lines so that in each corner there is a square of order n/4 and in the center a square of order n/2.

3. Within these 5 squares, interchange all pairs of numbers symmetrically opposite the square’s center. Outside the 5 squares, leave the numbers as they are.

The results are shown in b of the 4-by-4 and 8-by-8 diagrams. Magic squares so formed are symmetric.

Two problems to be done on your own:

In forming a magic square of order 4ka, we reverse step 3. We leave the numbers in the 5 squares as is. In the remaining four rectangles we interchange all pairs of numbers symmetrically opposite the square’s center. Result: a magic square.

Form a magic square of order 12.

Squares of even orders that are not multiples of 4: To make magic squares of orders 6, 10, 14, 18,..., one of the best methods is to put a frame around a magic square of order 4n, as shown here. Within the original square (here the order-4 square we made before) each number is raised by (2n – 2), where n is the order of the square we wish to form (here, 6). In this case, 1 becomes 1 + 10 = 11, 2 becomes 12, 3 becomes 13, and so on. The new order-4 square is a in the last diagram. It is always possible to place 1 through 10 and 27 through 36 (see b) so that the result is a magic square with magic constant (n3 + n)/2. Here n = 6, so the constant is 111.

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