Unit 3: Problem Solving Across Cultures – Magic Squares
Learning Objectives
By the end of this unit, participants will be able to:
- Define a magic square and describe its mathematical properties.
- Identify examples of magic squares from at least two different cultural traditions.
- Explain the symbolic or cultural significance of magic squares in various historical contexts.
Introduction
Previously we have viewed a few different number systems, examining the cultural roots which led to their development as well as the qualities or characteristics of those systems which are still reflected in the mathematics we perform today.
Now, we will look into some problem-solving strategies employed by various cultures. To do this, we will examine a mathematical puzzle which has appeared over centuries in different cultures around the world: Magic Squares.
What are Magic Squares?
Magic squares are square grids containing numbers such that the sums of the number in each row, column, and the two diagonals are all equal.
The3 × 3 arrangement for a magic square is known as a Normal Magic Square, but any 𝑛 × 𝑛 magic square can be created using each of the numbers 1 through 𝑛2.
Notice how for the case of the 3 × 3 magic square, the sum for each column, row, and diagonal is 15. This is referred to as the magic number (or magic constant) for this type of magic square ( 3 × 3)
To see how the number of potential solutions increases as we increase the size of the magic square, read this discussion from Alison Eves exploring some interesting characteristics of larger magic squares. Additionally, she reveals a connection she found between magic squares that appear on Albrecht Dürer's etching of Melencolia I and the facade of the Sagrada Familia in Barcelona.
Magic Squares Across Cultures
It is very interesting that a rotated version of Dürer's magic square would be nearly identical to the one that appears on the Sagrada Familia.
Could this be because the creators of each shared similar values, interests, and beliefs? Or could this simply be a coincidence caused by the symmetrical properties of squares?
Regardless of the reason, it is worth exploring some of the cultural significance embedded in magic squares.
Consider the origins of the first recorded magic square, the Lo Shu Square, found in ancient China roughly 5000 years ago.
Michael Daniels in his TedTalk, "The magic, myth and math of magic squares" explores some of the cultural significance of this square, highlighting the placements of even numbers in each of the four corners and odd numbers to form a cross in the middle of the square. In Chinese mythology, even numbers are referred to as the "yin" or "passive" numbers whereas odd numbers are referred to as the "yang" or "active" numbers, and the cross they form has significance in feng shui.
Additionally, tracing a path following the numbers in order reveals a pattern very similar to movements known as the Steps of Yu, a pattern of motion used by Taoist priests when moving through temples.
It can be seen that the placement of the numbers in this magic square greatly reflects qualities of beliefs of the culture it came from.
Watch Michael Daniels' TedTalk to see more ways in which magic squares reflect the cultures they originated in, as well as to be introduced to some fascinating connections
between a magic square found in the Jain temple of northern India, the magic square of Dürer's Melencolia I (which we saw earlier), and the planetary squares of Cornelius Agrippa.
Create Your Own Magic Square
Try to create a magic square of your own desire! This square can be as big or as small as you want (can't be 1 × 1 or 2 × 2), but remember that all of the sums of numbers in each of the rows, columns, and the two diagonals must equal the same value.
Try to consider how the placement of numbers in the square can reflect values or interests from everyday life, or maybe about how you simply prefer a certain layout (or pattern) for the numbers in the square.
Take some time to draw the square on a sheet of paper or feel free to use software (Microsoft word, Draw.io, etc) to create a square.
Does your magic square reflect some of your interests or values which you encounter on an everyday basis? Why or why not?
What is one example of a magic square reflecting qualities of the culture it originated from?