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![Page 1: Using Magic Squares to Study Algebraic Structure Bret Rickman MS, M.Ed. Portland State University Portland Community College “I have often admired the.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d305503460f94a09576/html5/thumbnails/1.jpg)
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)
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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.
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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!
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What are Magic Squares?
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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
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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:
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The Myth – Emperor Yu & Lo-Shu
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Chinese Emperor Yu 2800 BCE (650 BCE) Myth of the turtle. Lo-Shu (scroll of the
river Lo).
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Theon of Smyrna Greek Philosopher &
Mathematician.
On Mathematics Useful for
the Understanding of Plato
(130 CE)
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Varahamihira Indian Mathematician and
Astronomer.
Perfume recipe using magic
square in Brhatsamhita,
around the year 550 CE.
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Leonard Euler
Legendary Swiss
Mathematician 1707-1783.
Found magic squares
“entertaining”.
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Magic Square Artwork
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Albrecht Durer German Artist &
Mathematician.
Melencolia I – Copper
Engraving (1514 CE)
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Melencolia I
Source: wisdomportal.com
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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
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Source: pballew.net
On display at Eaton Fine Art Gallery in West Palm Beach, FloridaOrder 3 : Magic Constant = 30.
Magic Square ArtworkPatrick Ireland
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Magic Square
Cool Math
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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:
![Page 19: Using Magic Squares to Study Algebraic Structure Bret Rickman MS, M.Ed. Portland State University Portland Community College “I have often admired the.](https://reader035.fdocuments.in/reader035/viewer/2022062221/56649d305503460f94a09576/html5/thumbnails/19.jpg)
Magic Square Other Configurations
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Other Configurations: Magic Triangles
Magic Sum = 9
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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.
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Magic Square
Technology
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Magic Square Technology – Using Spreadsheets
AddingMagicSquares
MultiplyMagicSquares
Verify Associative Property of Addition
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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).
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Magic Square Curriculum Piece
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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
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Skill Practice
16 2 12 6 10 148 18 4
30
30
30
30
30 30 30 30
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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
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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
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Math Operation Magic!
Scalar Addition, Subtraction, Multiplication & Division
Activity Sheet # 3
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More Math Operation Magic!
Magic Square Addition & Grouping
Activity Sheet # 4
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Advanced Math Operation Magic!
Magic Square Matrix Multiplication
Activity Sheet # 5
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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
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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.
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Audience Questions
Any questions that you might have about magic squares or this curriculum are welcomed and encouraged!
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Have fun with Magic Squares. You’re in good company!Thank you for your attendance and participation.