The Mathemagic of Magic Squares History of Magic Squares - - PowerPoint PPT Presentation

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The Mathemagic of Magic Squares History of Magic Squares - - PowerPoint PPT Presentation

May 31, 2023 88 likes •969 views

The Mathemagic of Magic Squares Steven Klee Outline What is a Magic Square? The Mathemagic of Magic Squares History of Magic Squares Mathematics and Magic Squares Constructing Steven Klee Magic Squares Magic Circles University of

slide-1
SLIDE 1

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The Mathemagic of Magic Squares

Steven Klee

University of California, Davis

April 15, 2012

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SLIDE 2

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.
slide-3
SLIDE 3

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

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SLIDE 4

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

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SLIDE 5

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

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SLIDE 6

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

slide-7
SLIDE 7

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

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SLIDE 8

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

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SLIDE 9

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

slide-10
SLIDE 10

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

slide-11
SLIDE 11

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up

The 15 Game Players take turns choosing numbers between 1 and 9, without

  • repeats. The first player to choose 3 numbers that add up to 15 wins.

1 2 3 4 5 6 7 8 9

Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1

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SLIDE 12

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

1

What is a Magic Square?

2

History of Magic Squares

3

Mathematics and Magic Squares

4

Constructing Magic Squares

5

Magic Circles

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SLIDE 13

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Definition

Definition A magic square is a filling of an n × n square with the numbers 1, 2, . . . , n2 so that the rows, columns, and diagonals all sum to the same number.

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SLIDE 14

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Definition

Definition A magic square is a filling of an n × n square with the numbers 1, 2, . . . , n2 so that the rows, columns, and diagonals all sum to the same number.

34 1 15 14 4 12 6 7 9 8 10 11 5 13 3 2 16 34

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SLIDE 15

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The Lo Shu Square

Lo Shu Square: ∼ 650 BCE Magic Sum 15 is the number of days in the 24 cycles of the Chinese solar year.

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SLIDE 16

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The Chautisa Yantra

Chautisa Yantra: Parshvanath Jain temple in Khajuraho, India (10th century) 7 12 1 14 2 13 8 11 16 3 10 5 9 6 15 4

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SLIDE 17

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

D¨ urer’s Square

Albrecht D¨ urer: Melencolia I (1514)

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SLIDE 18

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Benjamin Franklin’s Squares

“The Governor put me into the commission of the Peace; the Corporation of the City chose me of the Common Council, and soon after an Alderman; and the Citizens at large chose me a Burgess to represent them in Assembly. This latter Station was the more agreeable to me, as I was at length tired with sitting there to hear Debates in which as Clerk I could take no part, and which were often so unentertaining, that I was induced to amuse myself with making magic squares, or circles, or anything to avoid weariness.”

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SLIDE 19

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Benjamin Franklin’s Magic Square

52 61 4 13 20 29 36 45 14 3 62 51 46 35 30 19 53 60 5 12 21 28 37 44 11 6 59 54 43 38 27 22 55 58 7 10 23 26 39 42 9 8 57 56 41 40 25 24 50 63 2 15 18 31 34 47 16 1 64 49 48 33 32 17

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SLIDE 20

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The Magic Sum

Question: What is the magic sum for an n × n magic square? ? ? ? · · · ? S ? ? ? · · · ? S ? ? ? · · · ? S ? ? ? · · · ? . . . ? ? ? · · · ? S n · S So n · S = 1 + 2 + 3 + · · · + n2

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SLIDE 21

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The Magic Sum

Question: What is the magic sum for an n × n magic square? ? ? ? · · · ? S ? ? ? · · · ? S ? ? ? · · · ? S ? ? ? · · · ? . . . ? ? ? · · · ? S n · S So n · S = 1 + 2 + 3 + · · · + n2 = n2(n2 + 1) 2

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SLIDE 22

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The Magic Sum

Question: What is the magic sum for an n × n magic square? ? ? ? · · · ? S ? ? ? · · · ? S ? ? ? · · · ? S ? ? ? · · · ? . . . ? ? ? · · · ? S n · S So n · S = 1 + 2 + 3 + · · · + n2 = n2(n2 + 1) 2 S = n(n2 + 1) 2

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SLIDE 23

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The Magic Sum

The Magic Sum The magic sum for an n × n magic square is n(n2 + 1) 2 . Example: n = 3 : S = 3 · (32 + 1) 2 = 3 · 10 2 = 15 n = 4 : S = 4 · (42 + 1) 2 = 4 · 17 2 = 34 n = 5 : S = 5 · (52 + 1) 2 = 5 · 26 2 = 65 n = 8 : S = 8 · (82 + 1) 2 = 8 · 65 2 = 260

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SLIDE 24

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.
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SLIDE 25

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections:

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SLIDE 26

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5

slide-27
SLIDE 27

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5

slide-28
SLIDE 28

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5 8 1 6 5

slide-29
SLIDE 29

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5 8 1 6 5

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SLIDE 30

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5 8 1 6 5

slide-31
SLIDE 31

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5 8 1 6 5 2

slide-32
SLIDE 32

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5 8 1 6 5 7 2

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SLIDE 33

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5 8 1 6 3 5 7 2

slide-34
SLIDE 34

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5 8 1 6 3 5 7 4 2

slide-35
SLIDE 35

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: 1 + 9 + 5 1 + 8 + 6 2 + 9 + 4 2 + 8 + 5 2 + 7 + 6 3 + 8 + 4 3 + 7 + 5 4 + 6 + 5 8 1 6 3 5 7 4 9 2

slide-36
SLIDE 36

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3 Player 2: 5 8 1 6 3 5 7 4 9 2

slide-37
SLIDE 37

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3 Player 2: 5 8 1 6 3 5 7 4 9 2

slide-38
SLIDE 38

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3 Player 2: 5 8 1 6 X 5 7 4 9 2

slide-39
SLIDE 39

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3 Player 2: 2 8 1 6 X 5 7 4 9 2

slide-40
SLIDE 40

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3 Player 2: 2 8 1 6 X 5 7 4 9 O

slide-41
SLIDE 41

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3, 6 Player 2: 2 8 1 6 X 5 7 4 9 O

slide-42
SLIDE 42

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3, 6 Player 2: 2 8 1 X X 5 7 4 9 O

slide-43
SLIDE 43

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3, 6 Player 2: 2, 5 8 1 X X 5 7 4 9 O

slide-44
SLIDE 44

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3, 6 Player 2: 2, 5 8 1 X X O 7 4 9 O

slide-45
SLIDE 45

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3, 6, 8 Player 2: 2, 5 X 1 X X O 7 4 9 O

slide-46
SLIDE 46

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3, 6, 8 Player 2: 2, 5, 1 X O X X O 7 4 9 O

slide-47
SLIDE 47

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The 15 game

Rules Two players take turns choosing numbers between 1 and 9. The

  • bjective is to collect three numbers that sum to 15.

Winning collections: Player 1: 3, 6, 8, 4 Player 2: 2, 5, 1 X O X X O 7 X 9 O

slide-48
SLIDE 48

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-49
SLIDE 49

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-50
SLIDE 50

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 → 2

slide-51
SLIDE 51

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 → 2

slide-52
SLIDE 52

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 → 4 9 2

slide-53
SLIDE 53

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 → 4 9 2

slide-54
SLIDE 54

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-55
SLIDE 55

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-56
SLIDE 56

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-57
SLIDE 57

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-58
SLIDE 58

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-59
SLIDE 59

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-60
SLIDE 60

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

8 1 6 3 5 7 4 9 2

slide-61
SLIDE 61

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Constructing Odd Magic Squares

1

Place 1 in the middle of the top row.

2

Having placed number i, place number i + 1:

1

One square to the northeast of i, if you can (wrapping if necessary).

2

One square to the south of i, otherwise.

17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9

slide-62
SLIDE 62

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

What about even Magic Squares?

When n = 2 · (2m + 1)

1

Start with a 2m + 1 × 2m + 1 magic square.

2

Fill another 2m + 1 × 2m + 1 square with the letters L, U, and X as follows:

slide-63
SLIDE 63

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

What about even Magic Squares?

When n = 2 · (2m + 1)

1

Start with a 2m + 1 × 2m + 1 magic square.

2

Fill another 2m + 1 × 2m + 1 square with the letters L, U, and X as follows:

1

Fill the first m + 1 rows with L.

2

Fill the next row with U.

3

Fill the remaining rows with X.

4

Replace the middle entry of the U row with the L above it.

8 1 6 3 5 7 4 9 2 L L L L U L U L U

slide-64
SLIDE 64

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The LUX Method

  • 3. Replace each square in the LUX grid with a 2 × 2 square

according to the rules: 2 4 1 3 2 3 1 4 3 1 2 4

slide-65
SLIDE 65

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The LUX Method

2 4 1 3 2 3 1 4 3 1 2 4 8 1 6 3 5 7 4 9 2 L L L L U L U L U 32 29 4 1 24 21 30 31 2 3 22 23 12 9 17 20 28 25 10 11 18 19 26 27 13 16 36 33 5 8 14 15 34 35 6 7

slide-66
SLIDE 66

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The LUX Method

2 4 1 3 2 3 1 4 3 1 2 4 8 1 6 3 5 7 4 9 2 L L L L U L U L U 32 29 4 1 24 21 30 31 2 3 22 23 12 9 17 20 28 25 10 11 18 19 26 27 13 16 36 33 5 8 14 15 34 35 6 7

slide-67
SLIDE 67

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

The LUX Method

2 4 1 3 2 3 1 4 3 1 2 4 8 1 6 3 5 7 4 9 2 L L L L U L U L U 32 29 4 1 24 21 30 31 2 3 22 23 12 9 17 20 28 25 10 11 18 19 26 27 13 16 36 33 5 8 14 15 34 35 6 7

slide-68
SLIDE 68

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Ben Franklin’s Magic Circles

“Dear Sir, As you seemed desirous

  • f seeing the magic circle I

mentioned to you, I have revised the one I made many years since, and with some improvements, sent it to you.” In a letter to John Canton, May 29, 1765.

slide-69
SLIDE 69

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Benjamin Franklin’s Magic Circle

3 8 1 7 5 6 2 1 2 2 1 6 7 6 5 2 3 7 2 1 4 6 1 2 7 6 8 1 8 1 6 7 4 8 3 1 5 7 4 6 4 7 5 6 5 1 4 9 3 9 3 2 5

12

7 4 1 3 7 5 2 6 6 2 2 6 4 1 5 7 3 5 8 4 5 4 3 5 2 3 4 5 4 3 2 4 2 3 5 5 3 3 3 5 5 2 9 5 9 3 6 2 8 3 7 4 1 4 4 4 2 6 6 1 9 6 9 7 1 2 4 6 3

slide-70
SLIDE 70

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Radial Sum

20

69 17 38 50 26 60 19 71 24 62

13 75

74 12 21 67 65 23 72 14 61 27 68 18 16 70 63 29 59 36 48 31 57 42 35 53 33 55 40 46 45 43 52 34 54 32 41 47 56 28 51 37 49 39 30 58 25 44 12

73 15 64 22 66

slide-71
SLIDE 71

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Outer-half Radial Sum

66 69 17 38 50 26 60 19 71 24 62 13 75 74 12 21 67 65 23 72 14 61 27 68 18 16 70 63 29 59 36 48 31 57 42 35 53 33 55 40 46 45 43 52 34 54 32 41 47 56 28 51 37 49 39 30 58 25 44 12

73 15 64 22

20

slide-72
SLIDE 72

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Inner-half Radial Sum

64 69 17 38 50 26 60 19 71 24 62

13 75

74 12 21 67 65 23 72 14 61 27 68 18 16 70 63 29 59 36 48 31 57 42 35 53 33 55 40 46 45 43 52 34 54 32 41 47 56 28 51 37 49 39 30 58 25 44 12 73 15 22

66 20

slide-73
SLIDE 73

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Annular Sum

18 69

17 38

50

26 60 19 71 24 62 13 75 74 12 67 65 23 72 14 61 27 68 16 70 63 29 59 36 48 31 57 42 35

53

33 55 40 46 45 43 52 54 32 41 47 56 28 51

37

49 39 30 58 25 44 12 73 15 22

66

20 64

21 34

slide-74
SLIDE 74

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Lower-half Annular Sum

18 69 17 38

50

26 60 19 71 24 62 13 75 74 12 67 65 23 72 14 61 27 68 16 70 63 29 59 36 48 31 57 42 35

53

33 55 40 46 45 43 52 54 32 41 47 56 28 51

37

49 39 30 58 25 44 12 73 15 22 66 20 64 21

34

slide-75
SLIDE 75

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Upper-half Annular Sum

18 69

17 38 50 26 60 19 71 24 62 13 75 74 12 67 65 23 72 14 61 27 68 16 70 63 29 59 36 48 31 57 42 35 53 33 55 40 46 45 43 52 54 32 41 47 56 28 51 37 49 39 30 58 25 44 12 73 15 22

66

20 64

21

34

slide-76
SLIDE 76

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

2×2 Block Sums

24

69 17 38 50 26 60 19

71

62 13 75 74 12 21 67 65 23 72 14 61 27 68 18 16 70 63 29 59 36 48 31 57 42 35 53 33 55 40 46 45 43 52 34 54 32 41 47 56 28 51 37 49 39 30 58 25 44 12 73

15 64

22 66 20

slide-77
SLIDE 77

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Northern Excentric Annular Sum

35

69

17

38

50

26 60 19 71 24 62 13 75 20 66 22 15 73 74 12 21 67 65 72 14 61 27 68 18 63 29 59 36 48 31 57 42 53 33 55 40 46 45 43 34 54 32 41 47 56 28 51 49 39 30 58 25 44 12 16

64 23 70 37 52

slide-78
SLIDE 78

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Eastern Excentric Annular Sum

16

69

17 38

50

26 60 19 71 24 62 13 75 20 66

22

64 15 73 74 12 21 67 65 72 14 61 27 68 18 70

63

29 59 36 48 31 57 42 35 53

33

55 40 46 45 43 52 34 54 41 47

56

28 51 37 49 39 30 58 25 44 12

32 23

slide-79
SLIDE 79

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Southern Excentric Annular Sum

48

69

17

38 50 26 60 19 71 24 62 13 75 20

66

22 64 15 73 74 12 67 65 23 72 14 61 27 68 18 16 63 29 59 36 31 57 42 35 53 33 55 46 45 43 52 34 54 32 41 56 28 51 37 49 30 58 25 44 12

70 21 39 47 40

slide-80
SLIDE 80

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Western Excentric Annular Sum

55

69 17 38 50 26 60 19 71 62 13 75 20 66 22 15 73 74 12 21 67

65

23 72 14 61 27 68

18

16 70 63 29 59 36 48 57 42 35 53 33 40 46 45 43 52 41 47 56 28 51 49 39 30 58 25 44 12 34 32

37 64 24 31 54

slide-81
SLIDE 81

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Vertically-centered Excentric Lower Half-annular Sum

35

69 17 38

50

26 60 19 71 24 62 13 75 20 66 22 15 73 74 12 21 67 65 72 14 61 27 68 18 63 29 59 36 48 31 57 42 53 33 55 40 46 45 43 34 54 32 41 47 56 28 51 49 39 30 58 25 44 12 16 64 23 70

37 52

slide-82
SLIDE 82

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Vertically-centered Excentric Upper Half-annular Sum

35 69

17

38 50 26 60 19 71 24 62 13 75 20 66 22 15 73 74 12 21 67 65 72 14 61 27 68 18 63 29 59 36 48 31 57 42 53 33 55 40 46 45 43 34 54 32 41 47 56 28 51 49 39 30 58 25 44 12 16

64 23 70

37 52

slide-83
SLIDE 83

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Horizontally-centered Excentric Right Half-annular Sum

16 69 17 38 50 26 60 19 71 24 62 13 75 20 66 22 64 15 73 74 12 21 67 65 72 14 61 27 68 18 70

63

29 59 36 48 31 57 42 35 53 33 55 40 46 45 43 52 34 54 41 47

56

28 51 37 49 39 30 58 25 44 12

32 23

slide-84
SLIDE 84

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Horizontally-centered Excentric Left Half-annular Sum

16

69

17 38

50

26 60 19 71 24 62 13 75 20 66

22

64 15 73 74 12 21 67 65 72 14 61 27 68 18 70 63 29 59 36 48 31 57 42 35 53

33

55 40 46 45 43 52 34 54 41 47 56 28 51 37 49 39 30 58 25 44 12 32 23

slide-85
SLIDE 85

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Benjamin Franklin

“The magic square and circle, I am told, have occasioned a good deal of puzzling among the mathematicians here, but no one has desired me to show him my method of disposing the numbers. It seems they wish rather to investigate it themselves.” In a letter to John Winthrop, July 2, 1768

slide-86
SLIDE 86

The Mathemagic

  • f Magic Squares

Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Thank you!