Using Magic Squares to Study Algebraic Structure Bret Rickman MS, M.Ed. Portland State University Portland Community College “I have often admired the mystical way of Pythagoras, and the secret magic of numbers.” Sir Thomas Browne ( )

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.

Examples of “Normal” Magic Squares  Normal Magic Square : Elements in order from rd Order Normal Magic Square th Order Normal Magic Square 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  Found magic squares “entertaining”.

Magic Square Artwork

Albrecht Durer  German Artist & Mathematician.  Melencolia I – Copper Engraving (1514 CE)

Melencolia I 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 Josep Maria Subirachs

Source: pballew.net On display at Eaton Fine Art Gallery in West Palm Beach, Florida Order 3 : Magic Constant = 30. Magic Square Artwork Patrick Ireland

Magic Square Cool Math

Examples of “Normal” Magic Squares  Normal Magic Square : Elements in order from rd Order Normal Magic Square th Order Normal Magic Square 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. Source: Harvey Heinz “Magic HyperCubes website.

Magic Square Technology

Magic Square Technology – Using Spreadsheets Adding Magic Squares Multiply Magic Squares Verify Associative Property of Addition

Magic Square Technology – Programming Bret’s “C” code  Magic Square Verification  Input 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

Skill Practice

Magic Square Creation Create 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 Method Link to method

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 Multiplication Is 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 Square Matrix 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.