## Leonhard Euler

## Magic Squares

Poyraz Ali Güner

The Basics of Magic Squares

Imagine waking up in a room, finding a plane with a 3x3 grid painted on it, balanced on a little marble with 45 cubes next to it. To get out, you must arrange the cubes in such an order that the sums of blocks in every line, colon, and diagonal in the grid should exactly be the same, causing the plane to be in balance.

The instructions above are more or less the way of creating what people call a Magic Square. These squares can have any size starting from a 3x3, consisting of numbers from 1 to where n is the length of a side of the square, and having all the sums mentioned above as equal.

Here is an example of the 3x3 square mentioned above. As you can see, it qualifies every condition counted above.

A common trait between every magical square is that they all have a magical sum. This is the sum of each line in the square. For instance, the magical sum for a 3x3 square is 15, while it is 34 for a 4x4 square. The method to calculate the magical sum for an n x n square is to divide the sum of numbers from 1 to by n. The sum of numbers from 1 to would be equal to

and since there are n of every line (except the diagonals; there are only 2 of them), it must be divided by n.

The Process of Making the Squares

There are three types of magic squares considered for their side lengths. The ones with an odd side length, a side length of a multiple of 4, and an even side length while not being a multiple of 4.

The first group is the easiest to handle. Here is a step-by-step example of a 3x3 square:

The rest of the steps also follow the rules above. Moreover, this procedure can be applied to any other magical square with an odd side length.

I won’t mention the process of constructing the squares with even side lengths, but more information can be found via the works cited.

The History of Magic Squares

The history of magic squares starts at approximately 2200 BC. According to a legend, a Chinese emperor named Yu discovered the image of this square on top of a nearby tortoise while he and his workers were building a riverbank on the Yellow River, which was known to be cursed.

After this legend, the earliest written records of magic squares are from 130 BC. Then, they were introduced to the Islamic world, and Arabic astronomers used them for their celestial maps. Afterwards, the squares spread towards Europe around the 15th century. They were first used in fortune-telling and alchemy, though they became an aspect of Western Africa in the 18th century and started to pop up in Probability problems and Calculus around the 19th century.

Uses of Magic Squares

Although magic squares are a rather abstract subject, these have some uses while balancing weights on a plate as I described in the beginning. These squares are also useful in Combinatorics, Geometry, and Graph Theory other than Probabilities and Calculus. Lastly, there is another version of magic squares called Latin squares, which are known as a variety of the game of Sudoku.

References

Cevahir, Ceyda. "Sihirli Kareler: Bir Kareden Çok Daha Fazlası" ["Magic Squares: More Than Just a Square"]. Matematiksel, Sept. 2017, www.matematiksel.org/aslinda-sadece-bir-kare-sihirli-karelerden-sudokuya-yolculuk/. Accessed 6 Aug. 2021.

Nesin, Ali. "Sihirli Kareler (I), (II)." Matematik Ve Korku, 2nd ed., İstanbul, Nesin Yayıncılık, 2010, pp. 157-79.

"Sihirli Kare" ["Magic Square"]. Vikipedi, https://tr.m.wikipedia.org/wiki/Sihirli_kare. Accessed 6 Aug. 2021.

Figure References

[1] “Sihirli Kare (Magic Square).” Wikipedia, Wikimedia Commons, 6 Apr. 2008, https://tr.m.wikipedia.org/wiki/Sihirli_kare.

[2] The figure is drawn by the author Poyraz Ali Güner.

Figure 1: A 3x3 Magic Square

Figure 2: The Steps to Form a 3x3 Magic Square