Using Magic Squares to Study Algebraic StructureBret Rickman MS, M.Ed.Portland State UniversityPortland Community College

“I have often admired the mystical way of Pythagoras, and the secret magic of numbers.”Sir Thomas Browne (1605-1682)

What to Expect Why Magic Squares? What are Magic Squares? Background history /artwork. Magic Square cool math. Activity – Constructing Magic Squares. Activity – Basic Operations, Matrix

Multiplication. Reflections on curriculum – further

explorations. Questions.

Why Magic Squares? Idea from Dr. Michael Mikusa (Kent State Univ).

Progressive approach – simple to more complex Underlying link to algebraic structure

Bret’s previous attempt to teach Magic Squares Not very successful – desire to approach in a

different manner Magic Squares inherent nature as intriguing

and fun, yet offer a great learning vehicle!

What are Magic Squares?

Some Basic Magic Square Terminology

Magic Square: a square array of numbers configured so

that the sum of the numbers is the same for each row,

column and both diagonals.

Normal Magic Square: Elements in order from

Magic Constant (sum): Numeric sum of each row,

column and diagonal in a magic square.

Normal square

Magic Square “Order”: The number of rows or columns.

21 n

2( 1)

2

n nS

Examples of “Normal” Magic Squares Normal Magic Square: Elements in order from

21 n

8 1 63 5 74 9 2

3rd Order Normal Magic Square

 4  14 15 19 7 6 125 11 10 816 2 3 13

4th Order Normal Magic Square

2( 1)

2

n nS

Magic Sum:

The Myth – Emperor Yu & Lo-Shu

Chinese Emperor Yu 2800 BCE (650 BCE) Myth of the turtle. Lo-Shu (scroll of the

river Lo).

Theon of Smyrna Greek Philosopher &

Mathematician.

On Mathematics Useful for

the Understanding of Plato

(130 CE)

Varahamihira Indian Mathematician and

Astronomer.

Perfume recipe using magic

square in Brhatsamhita,

around the year 550 CE.

Leonard Euler

Legendary Swiss

Mathematician 1707-1783.

Found magic squares

“entertaining”.

Magic Square Artwork

Albrecht Durer German Artist &

Mathematician.

Melencolia I – Copper

Engraving (1514 CE)

Melencolia I

Source: wisdomportal.com

Source: wisdomportal.com

Passion Façade of Familia Sagrada: Holy Family Church- Barcelona, Spain

Magic Square Artwork

The magic constant of the square is 33, the age of Jesus at the time of the Passion.

Antoni Gaudi - 1915

Josep Maria Subirachs - 1987

Source: pballew.net

On display at Eaton Fine Art Gallery in West Palm Beach, FloridaOrder 3 : Magic Constant = 30.

Magic Square ArtworkPatrick Ireland

Magic Square

Cool Math

Examples of “Normal” Magic Squares Normal Magic Square: Elements in order from

21 n

8 1 63 5 74 9 2

3rd Order Normal Magic Square

 4  14 15 19 7 6 125 11 10 816 2 3 13

4th Order Normal Magic Square

2( 1)

2

n nS

Magic Sum:

Magic Square Other Configurations

Other Configurations: Magic Triangles

Magic Sum = 9

Other Configurations: Magic Cubes

There are rows, columns and pillars in a magic cube. All are required to sum to the magic constant.

There are 4 triagonals. All 4 must sum to the correct constant.These are the minimum requirements for a simple magic cube.

There may be some diagonals that sum correctly, but that is not a requirement for a simple magic cube.

23m

Source: Harvey Heinz “Magic HyperCubes website.

Magic Square

Technology

Magic Square Technology – Using Spreadsheets

AddingMagicSquares

MultiplyMagicSquares

Verify Associative Property of Addition

Magic Square Technology – Programming

Bret’s “C” code

Magic Square VerificationInput proposed array (of any “order”).Program determines its “magic-ness”.

Magic Square Generator (limited edition – 3x3 only)Generates all 9! permutations

(362,880) of which only 8 are magic (only one unique; no rotations / reflections allowed).

Magic Square Curriculum Piece

Skill Practice

Study the square on your activity sheet.

What is its magic constant?

Answer the remaining questions and stop when you’ve finished filling in

this square.

16 2 12 

18

Skill Practice

16 2 12 6 10 148 18 4

30

30

30

30

30 30 30 30

Magic Square CreationCreate your own Magic Squares!Must begin with an arithmetic sequence and be an “odd order” square.

Starting from the central box of the first row with lowest number in sequence.The fundamental movement for filling the boxes is diagonally up and right.

When a move would leave the square, it is wrapped around to the next row up (first column) or next column to the right (last row), respectively.

If a filled box is encountered, move vertically down one box instead, then continuing as before.

De La Loubere / Hindu / Staircase MethodLink to method

         

         

         

         

       

6

2

3

4

5 7

1 8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

5 x 5 Staircase Construction Method Animation

Math Operation Magic!

Scalar Addition, Subtraction, Multiplication & Division

Activity Sheet # 3

More Math Operation Magic!

Magic Square Addition & Grouping

Activity Sheet # 4

Advanced Math Operation Magic!

Magic Square Matrix Multiplication

Activity Sheet # 5

Magic Square Matrix MultiplicationIs matrix multiplication closed for magic squares?

What did you notice about the resulting square?

Can you make a conjecture about magic square matrix multiplication?

What about Magic SquareMatrix Multiplication Associativity?

Activity Sheet # 5

Reflections Fun curriculum to teach – great vehicle for

algebraic structure.

Proof of Staircase construction method would be a nice extension.

Proof of why matrix multiplication is closed only for semi-magic squares.

Need more technology integration for curriculum.

Audience Questions

Any questions that you might have about magic squares or this curriculum are welcomed and encouraged!

Have fun with Magic Squares. You’re in good company!Thank you for your attendance and participation.