Teaching Resource: The History of Squares

Kate Di Lallo

1. Introduction to Magic Squares 1.1 What are Magic Squares? A magic square is an N x N array of numbers, by which horizontal, vertical and diagonal (coming from each corner of the array) add up to the same number, which is labeled the ‘’. 20 The smallest magic square able to be made is a 3 x 3 array, or an ‘order 3’ magic square.19 Below features the simplest magic square, with the array featuring numbers from 1 to N2.14

Figure 1. Order 3 Magic Square Magic squares present the more ‘beautiful’ side of mathematics, and as such, these arrays have continued to fascinate through their interesting patterns throughout history.14 Magic squares have a rich history, believed to date back to 2200 B.C., 19 however, discoveries related to these arrays surfaced time and time again throughout history, as there seems to be endless discoveries to be made, whether it is new methods to construct the squares, formulas to find the magic constants, and application to shapes beyond the usual square.14 For the purpose of this activity, Chinese and Indian History will be considered.

1.2 History of Magic Squares 1.2.1 Chinese History The earliest record of Magic Squares is debated among researchers; it is suggested that records of Magic Squares can be dated back as far as 2200 B.C. in ancient , during the time where Emperor Yu was in power.14 According to Chinese legend, the Emperor discovered the number patterns whilst walking along the Yellow River,2 which is suggested to have been central to Chinese civilization in history, in

terms of its cultural implications.21 It is said that the Emperor saw a tortoise with markings on its back that intrigued him (see Figure 2. & 3.).

Figure 2. Lo Shu Tortoise8 Figure 3. Lo Shu Magic Square13

The Emperor noticed that the markings on the tortoise’s back demonstrated different number patterns; first he noticed that the number of markings on opposite cells on the shell added to 10, and that there was 15 markings in three cells in a column, row or diagonal.14 From this moment on, the tortoise had celebrity status, and the ancient markings (Figure 3) became well known as the Lo Shu Square14 (the plausibility of this legend would be an interesting topic of discussion with students).

The Lo Shu magic square is central in the ancient art of Feng Shui, which involves balancing different energies.18 In the , odd and even numbers alternate around the central number 5.17 It is suggested that the odd numbers were associated with yang, or the ‘male principle’, and the even numbers were associated with yin, or the ‘female principle’, all of which are centered, in balance, around the number 5.14 There are several applications of the Lo Shu Square in Feng Shui.

1.2.2 Indian History It is suspected that after magic squares’ initial discovery in China, that they then made their way into Indian mathematics.16 was far slower to recognize order 3 magic squares (900 AD), however, they are believed to be the first to discover a fourth order magic square (1100 AD). 14 This magic square is found in the Jain Temples, in Khajuraho, India (see Figure 4. & 5.) hence demonstrating magic squares’ religious implications

Figure 4. Magic Square at Jain Temples1 Figure 5. Translated Jain temple Magic square16

Much like China’s belief in the squares’ power via implications for Feng Shui, India interestingly also saw beyond just the magic squares’ mathematical implications, and too believed in the ‘magic’ of the squares2. For example: • The traditional order 3 magic square (i.e. Like the Lo Shu Square) is believed to help find missing people.14 • Travelers were believed to get safe passage with a order 4 square with a magic constant of 34.2 • Crying babies are said to be eased via an order 4 magic square, with a magic constant of 84. 2 • Other magic squares had the purpose of curing illness, revenge, reducing labour pains14 and procedures to predict the future.16

Moving forward to more modern accounts of the magic square, famous Indian mathematician, (1887-1920), communicated his insights on the construction of the number arrays.7 Ramanujan’s contributions in the area of mathematics were prolific, despite facing several adversities including severe poverty, a lack of educational resources, including no tertiary education.7 While his works were extensive, he did, indeed, make contribution to the concept magic squares, as documented in Chapter 1 of ‘Ramanujan’s Notebooks Part 1’, in particular, his methods of constructing magic squares of different characteristics.7 He too, is well known for what is called ‘Ramanujan’s Magic Square’, which is a 4 x 4 magic square, with a magic constant of 139.11 The patterns of numbers within this square exceed those of a regular magic square, and what is most

fascinating of all, is that Ramanujan’s birthday is 22 of December 1887: 22 + 12 + 18 +87 =139.

Figure 6. Ramanujan’s Magic Square 12

2. Teaching Resource 2.1 Lesson Instructions to Teacher The following resource is a worksheet, which contains two activities. The first pertains to Chinese history, and the second pertains to Indian History. During the first activity, students should be instructed to work within their small groups to investigate each part of the question. As a teacher, you can read the historical context as a class first. Give students sufficient time to answer the problems in their groups, and then discuss answers as a class. The second part of the activity involves a game, run by the teacher. The historical background at the beginning of the activity should be read as a class, as it explains the context for the game: as per Indian beliefs, students must administer you, the teacher with the magic square that you require for your needs. You must first say your need (Round 1: Missing child, Round 2: Safe Passage, Round 3: Crying Baby), and the students must analyze the three magic squares to administer you the correct one. The first group to raise their hand with the correct square wins the round. The team with the most points at the end of the three rounds wins.

2.2 Worksheet

A History of Magic Squares

1. In 2200 B.C, in ancient China, Emperor Yu was walking beside the Yellow River, minding his own business. By the water, he notices a tortoise - but this isn’t a regular looking tortoise. The tortoise’s shell seemed to be broken up into 9 different clusters of markings like seen below.

a) Write down the number of markings in each cluster in the respective cell in the table.

b) What Emperor Yu found on the tortoise’s shell, and what you have written in the table, is the very first magic square ever discovered, called the Lo Shu Square. In your groups, discuss why these squares are considered to be ‘magic’. Record your ideas below.

Hint: Consider the rows, columns, and diagonals… ______

c) What is the ‘magic constant’? ______

d) The Lo Shu Square is a central part of Chinese cultures’ belief in Feng Shui, which is a philosophical system relating to the balance of energy. By looking at the nature of the numbers, and their position in the array, think why this may be the case. ______

e) Fill in the blank cells to complete the magic squares. State the magic constant if necessary. i) ii)

11 14 11

12 20 22

2 21 30

Magic Constant = 48 Magic Constant =______

iii) iv)

12 4 42 4 10 23 9 5 41 33 14 22 40 12

13 16 20 3 35

Magic Constant= 52 Magic Constant=______

2. Even though they weren’t as quick as ancient Chinese mathematicians, Magic Squares were also prevalent in ancient Indian mathematics. They too also believe in the magic nature of the squares, however, they used them as a prescription, more or less, to get some kind of response. For example, some magic squares were believed to cure sickness, some were believed to stop pain, etc. The magic constant and the order (i.e. how many rows/columns it had) influenced the squares powers.

Your job: Mr./Mrs./Ms. ______requires the mysterious effects of one of the magic squares, but which one?!

Depending on Mr./Mrs./Ms. ______needs, you will need to give him/her the correct magic square.

Need Magic Square Characteristics Need help finding their missing Order 3 child. Magic constant of 15 Need help to stop their baby Order 4 from crying. Magic constant of 84 Need to be granted safe passage Order 4 on a long journey. Magic constant of 34

1. Need ______9 2 7 8 1 6 10 3 8 4 6 8 3 5 7 5 7 9 5 10 3 4 9 2 6 11 4

2. Need ______

5 17 12 6 16 2 3 13 16 6 12 1 11 7 4 18 5 11 10 8 4 9 7 15 8 14 15 3 9 7 6 12 5 17 2 11 16 2 9 13 4 14 15 1 10 3 14 8

3. Need ______

14 26 21 15 13 28 16 23 19 27 16 22 25 11 18 22 18 21 15 26 14 24 17 29 17 23 24 12 24 17 27 12 26 20 23 15 20 16 13 27 25 14 22 19 25 13 28 18

3. Rationale A constructivist approach to teaching is adopted via this activity, particularly in part 1, whereby the activity is largely learner-centred, in that students deduce the mathematical significance of the magic squares themselves. Specifically, inquiry based learning is facilitated, as students work collaboratively to investigate the relationship between numbers in the squares without teacher input.10 Both parts of the activity involve students considering a historical perspective of magic squares. A historical perspective of mathematics, indeed, has the potential to enhance student interest in the topic, as well as giving students a clear indication of the relevance and application of such concepts in real life.15

Within part 2 of the activity, students are asked to adopt the identity of a member of society in ancient India, and hence use magic squares as a means to evoke a certain outcome. This approach integrates drama-based instruction, as students are asked to immerse themselves in the problem, as if it were real life. In doing so, students are able to see the real life applications and historical significance of the concepts considered in the activity, which is found to lead to greater achievement (compared to traditional methods of teaching), greater retention, increased motivation, and an overall more positive attitude towards the topic.9 As such favourable outcomes are produced via a consideration of this teaching instruction, it was adopted in the activity.

4. Curriculum Links and Teaching Notes 4.1 Teaching Context While magic squares have the potential to be used as a tool in upper school mathematics teaching, the following teaching resource is designed for a lower school mathematics class, specifically a class of year 7 students. Students desks must be arranged in a way that allows for team work and collaboration, therefore small groups of desks is preferred.

4.2 Curriculum Links While it is suggested that this aspect of mathematics is considered to be largely ‘recreational’2, the historical context of magic squares has immense potential to be used as a tool to meet curriculum objectives, and hence a teaching resource has been designed to do so.

4.2.1 Mathematics Curriculum The teaching resource aims to meet several curriculum proficiencies expected of year 7’s in the subject of mathematics. The Australian Curriculum states that, students must be able to “Compare, order, add and subtract integers” (ACMNA280). 6 Indeed, the activity successfully allows students to demonstrate this objective as students are asked to investigate what makes magic squares ‘magic’ (i.e. discovering that triples in the squares add to the same number). Part e of question 1 also, indirectly, satisfies objectives pertaining to year 7 level algebra. Since students have to find the missing square, they are concurrently developing their algebra skills, specifically, solving equations.

Additionally, students must demonstrate the proficiency of problem solving by “formulating and solving authentic problems using numbers and measurements, working with transformations and identifying symmetry”.6 The students will be required to solve authentic problems using number, as they are founded upon historical maths discoveries and events. Students will be encouraged to acknowledge the symmetrical nature of the Lo Shu Square (meaning that cells on opposite sides add to 10, e.g. Top right and bottom left corner (8+2=10), middle cell in the top row and middle cell in the bottom row (9+1=10)). Finally, students will be given the opportunity to demonstrate the proficiency of fluency by “calculating accurately with integers”, 6 which, of course, is involved in finding the magic constants of the squares.

4.2.2 General Capabilities Indeed, the designed teaching resource allows students the opportunity to demonstrate several of the Australian Curriculum’s general capabilities, in an attempt to prepare students for challenges out of school contexts. • Literacy: Written components in the teaching resource (i.e. historical context). • Numeracy: Discerning the mathematical concepts behind magic squares. • Critical and Creative Thinking: Resource founded upon problem solving and inquiry. • Personal and Social Capability: The activity involves students working collaboratively. • Intercultural Understanding: The history of other cultures is considered, uncovering traditional beliefs and customs.4

4.2.3 Cross-curriculum Priorities Through the consideration of the history of magic squares within the subject of mathematics, the cross-curriculum priority of Asia and Australia’s Engagement with Asia.3 The Australian Curriculum suggest that the fulfillment of this priority is essential, due to the highly dependable relationship that Asia and Australia have, and as such, knowledge of Asian culture is vital for future workers. Indeed, the resource allows insight into two Asian cultures (China and India) from a historical context.

Additionally, the source lends itself to supporting an integrated curriculum. In teaching mathematics with a consideration of historical cultures, HASS curriculum objectives are achieved in conjunction with mathematics objectives. HASS students are required to learn about ‘The Ancient World’, which is inclusive of the two cultures considered in the activity.5

5. Enrichment Activity For students that require further enrichment beyond the mathematics involved in the teaching resource, students should be encouraged to consider the works of Srinivasa Ramanujan, specifically, his contributions in the area of magic squares. A potential activity is recommended below. Additionally (as included in the teacher recommended readings), teachers could consider including Ramanujan’s work as outlined in ‘Ramanujan’s Notebooks Part 1’, 7 specifically, the construction of magic squares with certain characteristics as a further enrichment idea, or alternatively, as a means of linking the topic of algebra to a real-life, historical application.

5.1 Enrichment Exercise Srinivasa Ramanujan (1887-1920) is a famous Indian mathematician. Ramanujan made extensive contributions in the area of mathematics, and interestingly, he too gave insights into magic squares and their construction. There is one magic square that Ramanujan is known for: it is a 4x4 magic square, with a magic constant of 139 (as seen below). As in the magic squares featured in the previous activity, the sum of all numbers in each row, in each column, and the diagonals equal to the magic constant of 139. However, Ramanujan’s design of this square features many more combinations of numbers that add to the magic constant…

By shading with different coloured pencils, show as many combinations of four numbers that add to this magic constant.

22 12 18 87

88 17 9 25

10 24 89 16

19 86 23 11

Fun Fact: Ramanujan’s Birthday is the 22nd of December 1887. Why is this significant?

6. References

1. Aczel, D.B. (2013). Sex, Math and Magic [Image]. Retrieved from https://www.huffingtonpost.com/amir-aczel/sex-math-and- magic_b_3444571.html

2. Anderson, D.L. (2001). Magic squares: Discovering their history and their magic. Mathematics teaching in the Middle School, 6(8), 466- 471. Retrieved from JSTOR scholarly archive.

3. Australian Curriculum. (n.d.a). Asia and Australia’s Engagement with Asia. Retrieved from https://www.australiancurriculum.edu.au/f-10- curriculum/cross-curriculum-priorities/asia-and-australia-s-engagement-with- asia/

4. Australian Curriculum. (n.d.b). General Capabilities. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/general-capabilities/

5. Australian Curriculum. (n.d.c). History. Retrieved from http://v7- 5.australiancurriculum.edu.au/humanities-and-social- sciences/history/curriculum/f-10?layout=1#level7

6. Australian Curriculum (n.d.d). Mathematics. Retrieved from https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/

7. Berndt, B.C. (1985). Ramanujan’s Notebooks Part 1. New York, N.Y.: Springer-Velag New York Inc.

8. Braatz-Brown, L. (n.d.). Lo Shu [Image]. Retrieved from http://mathforum.org/alejandre/magic.square/loshu.html

9. Duatepe-Paksu, A., & Ubuz, B. (2009). Effects of drama-based geometry instruction on student achievement, attitudes, and thinking levels. The Journal of Educational Research, 102(4), 272-286, 320.

10. Duchesne, S., & McMaugh, A. (2016). Educational Psychology (5th ed.). Melbourne: Cengage Learning.

11. Manattu, A.J. (2016). What is Srinivasa Ramanujan’s magic square? Retrieved from https://www.quora.com/What-is-Srinivasa-Ramanujans- magic-square

12. Manattu, A.J. (2016). What is Srinivasa Ramanujan’s magic square? [Image] Retrieved from https://www.quora.com/What-is-Srinivasa-Ramanujans- magic-square

13. Mastin, L. (n.d.). [Image]. Retrieved from http://www.storyofmathematics.com/chinese.html

14. Pickover, C.A. (2011). The zen of magic squares, circles, and stars: an exhibition of surprising structures across dimensions [Princeton University Press]. Retrieved from EBSCOhost eBooks academic collection.

15. Savizi, B. (2007). Applicable problems in the : Practical examples for the classroom. Teaching Mathematics and its Applications, 26(1), 45-50. Retrieved from EBSCOhost.

16. Tanega, I.J. (2010). Equivalent mersions of “Khajuraho” and “Lo Shu” magic squares and the day 1st October 2010 (0.1.10.2010). Retrieved from https://arxiv.org/pdf/1011.0451.pdf

17. Tchi, R. (2017a). The Feng Shui magic of the LoShu Square. Retrieved from https://www.thespruce.com/feng-shui-magic-of-the-lo-shu-square-1274879

18. Tchi, R. (2017b). What is Feng Shui?. Retrieved from https://www.thespruce.com/what-is-feng-shui-1275060

19. University of Cambridge. (n.d.). An introduction to magic squares. Retrieved from https://nrich.maths.org/2476

20. Weisstein, E.W. (2017). Magic squares. Retrieved from http://mathworld.wolfram.com/MagicSquare.html

21. Wu, A. (2017). The Yellow River- “Mother river of China”. Retrieved from https://www.chinahighlights.com/yellowriver/

7. Recommended Readings 7.1 For Students

Pickover, C.A. (2011). The zen of magic squares, circles, and stars: an exhibition of surprising structures across dimensions [Princeton University Press]. Retrieved from EBSCOhost eBooks academic collection.

7.2 For Teachers

Berndt, B.C. (1985). Ramanujan’s Notebooks Part 1. New York, N.Y.: Springer-Velag New York Inc. (Specifically, Chapter 1: Magic Squares).