Magic Boxes

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Definition

A Magic Square is a two-dimensional array of numbers in which the sum of all rows, columns, and diagonals equal the same. This sum is called a magic number.

Background

The time frame, which Magic Squares, were first introduced is under much debate. In ancient times, these Magic Squares were used for different purposes. They possessed mystical and magical powers and some even served as good luck charms. For example, a Magic Square engraved on a silver platter was used as a charm against a plague.

College Essays on Magic Boxes

The first or earliest Magic Square I found was the Chinese Magic Square also known as the "lo-shu". This one dates back to roughly around 500 B.C. and measured x square. Another Magic Square that is well known in mathematics is the Albrecht Durers Melancholia, this 4x4 square indicates the date of its creation in the last row, 15 14.

The Lo-Shu Legend

According to material that I have read regarding Magic Squares, there is a story that tries to explain the Lo-Shu square and it goes like this

In the ancient time of China, there was a huge flood. The people tried to offer some sacrifice to the 'river god' of one of the flooding rivers, the 'Lo' river, to calm his anger. However, every time a turtle came from the river and walked around the sacrifice. The river god didn't accept the sacrifice until one time; a child noticed a curious figure on the turtle shell. Hence they realized the correct amount of sacrifices to make (15).

Now according to research, the number 15 is a magic number for a x magic square, but at this time it is not known what the turtle shell looked like. So from this story, the story of the Emperor Yu was created.

Emperor Yu was standing at the shore of the mighty Huang-He (Yellow River) thinking about his day. It was soon going to be evening and this was good, because the Emperor had a long and difficult day.

While the Emperor looked upon the river his dealing of the day seemed to slip away with the rush of the water. As he gazed on, he found the rivers edge was right below his feet. It was at this moment Emperor Yu saw the divine turtle. The Emperor had seen the turtle, but only as a pattern in the stars, never this close. Emperor Yu knew the story of Lo-Shu and believed that this turtle was a symbol of good luck.

The Emperor approached the turtle in an effort to get a better look at the creature. The Emperor was familiar with its shape, but the design on the shell was new to him. As he picked up the turtle he noticed that the shell looked like puzzle pieces glued together to form two circles around a rectangle. He looked long at these shapes and noticed a pattern of dots etched on them.

Starting with the right leg of the turtle he found that there was a square connected by four dots. Now moving clock wise he saw a row of nine dots. At the five o'clock position, there were two dots. At the bottom or the six o'clock position there were a row of seven linked dots. Continuing, he saw another box with six connected dots, one single dot, another box with eight dots, and at the top a row of three connected dots. In the center of all these he saw a cross-section of five dots.

After a long time pondering these numbers he lined them up and added up the numbers in many different ways.

His sum of the numbers was 15. But what did this mean? What did the sum have to do with him? Was it years of good luck? Or was it years to live? All the questions came easy to him but the answers were no where to be found.

Emperor Yu had come to the river for peace after a hard day, but instead found himself confused and uncertain.

Now let's see how we can create a magic square!

Construction

For me, the easiest magic square construction to explain is the 4x4 magic square. First you would create the square and number the boxes from 1 to 16 (1 to n).

1 4

5 6 7 8

10 11 1

1 14 15 16

Next, you mark the diagonals as highlighted. Take the sum of the first and last numbers of the consecutive ordering (1+16 = 17; or 1 + n). Then, change the other numbers as follows Take the difference between 17 and x, where x equals the number in the cell to be changed. The new numbers are

This is how the new square looks, and the sum is 4.

1 15 14 4

1 6 7

8 10 11 5

1 16

Conclusion

Magic Squares do not end with 4x4 arrays; they can grow as large as you want. However, the difficulty is much greater. While writing this paper and doing research, I came to realize the reason for such interest of these squares is that mathematicians are always looking for new information on numbers. They are always looking at new possibilities to calculate numbers and hopefully find something that no one has yet discovered.

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