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

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 California, Davis April 15, 2012

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

Warm-Up The Mathemagic of Magic Squares The 15 Game Steven Klee Players take turns choosing numbers between 1 and 9, without Outline repeats. The first player to choose 3 numbers that add up to 15 wins. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 1 2 3 4 5 6 7 8 9 Constructing Magic Squares Magic Circles Player 1: Player 2: 3, 6, 8, 4 2, 5, 1

The Mathemagic of Magic Squares Steven Klee What is a Magic Square? Outline 1 What is a Magic Square? History of Magic History of Magic Squares 2 Squares Mathematics and Magic Squares Constructing Mathematics and Magic Squares 3 Magic Squares Magic Circles Constructing Magic Squares 4 Magic Circles 5

Definition The Mathemagic Definition of Magic Squares Steven Klee A magic square is a filling of an n × n square with the numbers 1 , 2 , . . . , n 2 so that the rows, columns, and diagonals all sum to the Outline same number. What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles

Definition The Mathemagic Definition of Magic Squares Steven Klee A magic square is a filling of an n × n square with the numbers 1 , 2 , . . . , n 2 so that the rows, columns, and diagonals all sum to the Outline same number. What is a Magic Square? History of Magic Squares 34 Mathematics and Magic Squares Constructing 1 15 14 4 Magic Squares Magic Circles 12 6 7 9 8 10 11 5 13 3 2 16 34

The Lo Shu Square The Mathemagic of Magic Squares Steven Klee Lo Shu Square: ∼ 650 BCE Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares Constructing Magic Squares Magic Circles Magic Sum 15 is the number of days in the 24 cycles of the Chinese solar year.

The Chautisa Yantra The Mathemagic Chautisa Yantra: Parshvanath Jain temple in Khajuraho, India of Magic Squares (10th century) Steven Klee Outline What is a Magic Square? History of Magic Squares Mathematics and Magic Squares 7 12 1 14 Constructing Magic Squares 2 13 8 11 Magic Circles 16 3 10 5 9 6 15 4

D¨ urer’s Square The Mathemagic of Magic Squares Albrecht D¨ urer: Melencolia I (1514) 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 Mathemagic of Magic Squares “The Governor put me into the commission of the Peace; the Steven Klee Corporation of the City chose me of the Common Council, and soon Outline after an Alderman; and the Citizens at large chose me a Burgess to What is a Magic Square? represent them in Assembly. History of Magic This latter Station was the Squares more agreeable to me, as I was Mathematics and Magic Squares at length tired with sitting Constructing there to hear Debates in which Magic Squares Magic Circles 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.”

Benjamin Franklin’s Magic Square The Mathemagic of Magic Squares Steven Klee 52 61 4 13 20 29 36 45 Outline What is a Magic 14 3 62 51 46 35 30 19 Square? History of Magic Squares 53 60 5 12 21 28 37 44 Mathematics and Magic Squares 11 6 59 54 43 38 27 22 Constructing Magic Squares Magic Circles 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

The Magic Sum Question: What is the magic sum for an n × n magic square? The Mathemagic of Magic Squares Steven Klee ? ? ? ? S · · · Outline ? ? ? ? S · · · What is a Magic Square? ? ? ? ? S · · · History of Magic Squares . . ? ? ? ? . Mathematics and · · · Magic Squares Constructing ? ? ? ? S · · · Magic Squares Magic Circles n · S So 1 + 2 + 3 + · · · + n 2 = n · S

The Magic Sum Question: What is the magic sum for an n × n magic square? The Mathemagic of Magic Squares Steven Klee ? ? ? ? S · · · Outline ? ? ? ? S · · · What is a Magic Square? ? ? ? ? S · · · History of Magic Squares . . ? ? ? ? . Mathematics and · · · Magic Squares Constructing ? ? ? ? S · · · Magic Squares Magic Circles n · S So 1 + 2 + 3 + · · · + n 2 = n · S n 2 ( n 2 + 1) = 2

The Magic Sum Question: What is the magic sum for an n × n magic square? The Mathemagic of Magic Squares Steven Klee ? ? ? ? S · · · Outline ? ? ? ? S · · · What is a Magic Square? ? ? ? ? S · · · History of Magic Squares . . ? ? ? ? . Mathematics and · · · Magic Squares Constructing ? ? ? ? S · · · Magic Squares Magic Circles n · S So 1 + 2 + 3 + · · · + n 2 = n · S n 2 ( n 2 + 1) = 2 n ( n 2 + 1) S = 2

The Magic Sum The Mathemagic of Magic Squares The Magic Sum Steven Klee The magic sum for an n × n magic square is Outline n ( n 2 + 1) What is a Magic Square? . 2 History of Magic Squares Example: Mathematics and Magic Squares Constructing Magic Squares S = 3 · (3 2 + 1) = 3 · 10 Magic Circles n = 3 : = 15 2 2 S = 4 · (4 2 + 1) = 4 · 17 n = 4 : = 34 2 2 S = 5 · (5 2 + 1) = 5 · 26 n = 5 : = 65 2 2 S = 8 · (8 2 + 1) = 8 · 65 n = 8 : = 260 2 2