Magic squares

Selected topic 2

Title

MAGIC SQUARES

Objectives:

After reading this module, you will be able to:1. form the 3 x 3 magic square by recalling the interesting characteristics of this order-3 square using the natural numbers 1 to 9;2. appreciate the prodigious efforts of mathematicians to systematize the formation of higher order magic squares;3. derive 3 x 3 magic square algebraically; and4. trace the unicursal path of the sequence of numbers on a magic squares.

History:

The first record of a magic square appeared more 4000 years ago in China. The first magic square was supposedly brought to man by a sacred turtle from the River Lo in the days the legendary Emperor Yii, reputed to be a hyraulic engineer (Boyer, 1991, p. 197). The completed square is the Lo Shu of ancient China.According to legend, the pattern was first revealed on the back of turtle in the twenty-third century BC,

History:but the modern Chinese scholars trace reference to it to the fourth century BC. From then on until the 10th century the pattern was a mystical Chinese symbol enormous significance. (Gardner, 1988; p 215). The even numbers were identified with yin, the female principle; and the odd numbers with yang, the male principle. The central 5 represented the earth around which, in evenly balanced yin and yang, were the other four elements; 4 and 9 symbolizing metal, 2 and 7, fire; 1 and 6, water; and 3 and 8, wood.

History:

4 9 2

3 5 7

8 2 6

earth

metal

fire

wood

waterFig. 1: The Magic Square of China

History:In the magic square, the numbers from 1 to 9 are arranged in a square so that the sum of the three numbers in each row, column and diagonal is the same. In the Fig. 1 the sum is always 15.The number 5 is always in the central cell. The even numbers are in the corner cell. If we black out the odd numbers, we get a figure. If we link the odd numbers, we get a Z in an oblique position. If we link the even numbers, the path traced is a Z.

History:

The movement traced 1, 2, 3 7,8, 9 and 1, 4, 7, are V forms

1, 3, 5, 7, 9

Even numbers

History:And the path formed by the four ( 7, 8, 9; 3, 6, 9; 1, 4, 7 and 1, 2, 3 sequences is:

4 9 2

3 5 7

8 1 6

History:The magic square has other interesting characteristics. Arithmetic sequences are formed by the three numbers in the middle row, in the central column, and in each diagonal.

3, 5, 7 : difference 2 middle row1, 5, 9 : difference 4 central row4, 5, 6 : difference 1 diagonal2, 5, 8 : difference 3 diagonal

? SAQ 8 -1

9 8

6

1. Supposed only three numbers are given in 3-by-3 magic square. How are you going to place the other six numbers using x and y?

2. Do you find arithetic sequence in the resulting magic square?

9 8

x 6 y

History:

..

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you can make your own 3-by-3 magic square by using the patterns that appear in every such square.

+

.

.

.

..

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. . ....

. . .+ =

T T T 3T

Magic Square

In the diagram, the three spots in the first squares indicate sets of numbers whose sum is the same number T, which is the sum of the three numbers in each row, column and diagonal of a 3-by-3 magic square. When the nine spots are transferred to a single square, their total is 3T. However, the spots in the top and bottom rows each add up to T. In other words, the number in the central cell is always one-third of T.

Magic Square

If the number in the central cell is m, then those in the diagonal must form a sequences m – a, m, m + a ; m – b, m, m + b, that have a sum of 3m = T. Hence, the algebraic pattern for every 3-by-3 magic square is

m + a m - a - b m + b

m - a + b m m + a - b

m - b m + a + b m - a

Magic Square

where m, a, b are any numbers integral of fractional

m + a m - a - b m + b

m - a + b m m + a - b

m - b m + a + b m - a

Question: What will happen if a = 2b?Which cells will be affected most?

Magic Square

To avoid repeated number in a magic a ≠ 2b and b ≠ 2a are necessary. And to avoid zero or negative numbers m must exceed (a + b). The smallest set of positive integers fulfilling these conditions is m = 5, a = 3, b= 1, and these produce the Chinese magic square made with the numbers 1 to 9.

Magic Square

A Standard magic square is a square array of positive integers from 1 to arranged so that the sum of every row, every column, and each of the two main diagonals is th same. N is the order of the square. It is easy to see that the magic constant is the sum of all numbers divided by N. the formula is

(1+2+3+…+𝑁 2)𝑁

=(𝑁 3+𝑁 )

2

Magic Square

The trivial square of order is simply the number 1; and of course, it is unique. It is equally trivial to prove that no order-2 square is possible.The order-3 magic square composed of the integers 1 to 9, when rotations and reflections are excluded, is unique. The constant sum is

This is

Magic Square

There are eight ways in which the sum 15 can be partitioned:9 + 5 + 19 + 4 + 28 + 6 + 18 + 5 + 28 + 4 + 37 + 6 + 27 + 5 + 36 + 5 + 4

These correspond to the six orthogonals (rows and columns) and the two diagonals.

Magic Square

The eight lines exactly match the number of triples we have available. Since the center number belongs to a row, a column and both diagonals, it clearly must be a digit that appears in four of the eight triples. The only such digit is 5. Hence, we know that 5 is the central number. The number 9 belongs to only two triples. We cannot place it in a corner because each corner cell belongs to three lines. Consequently, it must go in a side cell.

Magic Square

Because of the square’s symmetry, it does not matter which cell we choose, so let us put it above 5. For the top corner on each side of 9, we have choices, so except 2 and 4. Again, it does not matter which digit goes where since one arrangement is merely a mirror reflection of the other. The rest of the square follows automatically. This simple instruction proves the uniqueness of this square.

FIVE-BY-FIVE-SQUARES

How many magic square are of order 5? According to Gardner (1991), the best estimate was given by Albert Candy in his Construction, Classification and Census of Magic Square of the Order Five, privately published in 1938. Candy arrived at a total of 13,288,952. The exact number was not known until 1973 when the counting was completed by a computer programmer at Information International. The program, using a standard backtracking procedure,


consists of about 3500 “words” and took about 100 hours of running time on a PDP-10. A final report was issued in 1975.Candy’s estimate was far below Schroeppel’s count. Not counting rotations and reflections, there are 245,305,224 magic square order of 5. Schroeppel prefers to divide that number 4 and give the total 68,826,306. The reason is that there are four other variants generated by the following two transformations which also preserve magic:


1. Exchange the left and right border columns, then exchange the top and bottom border rows.

2. Exchange rows 1 and 2 and rows 4 and 5. then exchange columns 1 and 2 and columns 4 and 5.

When the two transformations are combined with the two reflections and four rotations, the result is 2 x 4 x 2 x 2 =32 forms that can be called isomorphic. With this definitions of isomorphic, the count becomes 68,826,306.


That number can be lowered even more by considering another well known transformation. If every number in a magic square is subtracted from (in this case 26), the result called the complement, is also magic. When the center of an order-5 square is 13, the complement is isimorphic with the original. If it is not 13, a difference square results. If we broaden the term isimorphic to include complements, the count of order-5 drops to about 35 million.


Here is a 5-by-5 magic square:

1 15 24 8 17

23 7 16 5 14

20 4 13 22 6

12 21 10 19 3

9 18 2 11 25


Consider these numbers of order-5 square whose centers are number 1 through 13.

Number at Center123456

No. of different Order-5 squares possible

1,091,4481,366,1791,914,9841,958,8372,431,8062,600,879


Number at Center789

10111213

No. of different Order-5 squares possible

3,016,8813,112,1613,472,5423,344,0343,933,8183,784,6184,769,936


Note that the totals steadily increase from 1 through8 but the totals for 9 and 10 and 11 and 12 are reversed in rank. This is a surprising result. These same trends occur in counts of squares with centers 14 through 25, since these are complements of the centers 1 to 12. There are as many squares with 1 in the centers as there are with 25; and the same is true for all numbers except 13 (which is the same as its complement).


The numbers of squares listed for center 1 through 13 add up to the earlier figure cited as Schroeppel’s calculation with the help of the computer.Do you think the study of the number of order-5 square could have been possible without the help of computers? Explain your answer.One of the major breakthrough in mathematics has been made in the study of magic square. A further development and breakthrough is on magic cubes. However, that is not within the scope of this module.

Magic squares

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Transcript of Magic squares