Office Applications and Entertainment, Magic Squares

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List of Attachments

Magic Squares (4 x 4):

Attachment 1

Magic Squares 4 x 4, MC =  34, All  Diagonals

Attachment 2.3.1

Magic Squares 4 x 4, MC =  34, Main Diagonals

Attachment 2.5

Magic Squares 4 x 4, MC =  34, Associated

Magic Squares (5 x 5):

Attachment 5.2.6

Magic  Squares 5 x 5, MC = 65, a(25) = 1, 2, ... 25

Attachment 3.4.1

Magic  Squares 5 x 5, MC = 65, Variables a(20) ... a(25) constant

Attachment 3.4.2

Magic  Squares 5 x 5, MC = 65, Diagonal  a( 1) ... a(25) constant

Attachment 3.7.1

Sudoku Squares 5 x 5, MC = 10, Pan Magic

Attachment 3.7.2

Sudoku Squares 5 x 5, MC = 10, Magic

Attachment 3.7.3

Sudoku Squares 5 x 5, MC = 10, Magic, Non Sudoku Diagonals

Attachment 5.3.4

Magic  Squares 5 x 5, MC = 65, Unique Base Squares

Attachment 5.4.1

Magic  Squares 5 x 5, MC = 65, Ultramagic Squares

Attachment 5.4.2

Magic  Squares 5 x 5, MC = 65, Associated Squares

Attachment 5.5.1

Magic  Squares 5 x 5, MC = 65, Concentric Squares (1)

Attachment 5.5.2

Magic  Squares 5 x 5, MC = 65, Concentric Squares (2)

Attachment 5.5.3

Magic  Squares 5 x 5, MC = 65, Eccentric  Squares (1)

Attachment 5.5.4

Magic  Squares 5 x 5, MC = 65, Eccentric  Squares (2)

Attachment 5.6.1

Magic  Squares 5 x 5, MC = 65, Diamond Inlay

Attachment 5.6.2

Magic  Squares 5 x 5, MC = 65, Diamond Inlay, Associated

Attachment 5.6.3

Magic  Squares 5 x 5, MC = 65, Diamond Inlay, Concentric

Attachment 5.6.4

Magic  Squares 5 x 5, MC = 65, Square Inlay

Attachment 5.6.5

Magic  Squares 5 x 5, MC = 65, Square and Diamond Inlay

Attachment 5.5.2

Magic  Squares 5 x 5, MC = 65, Rotation/Reflection

Attachment 5.5.3

Magic  Squares 5 x 5, MC = 65, Line Permutations

Attachment 5.5.5

Magic  Squares 5 x 5, MC = 65, Class {Aijk}


Magic Squares (6 x 6)
General:

Attachment 6.5.1

Magic Squares  6 x 6, MC = 111, Highlighted Variables Constant (1)

Attachment 6.6.1

Magic Squares  6 x 6, MC = 111, Highlighted Variables Constant (2)

Attachment 6.5.2

Magic Squares  6 x 6, MC = 111, Rotation/Reflection

Attachment 6.5.3

Magic Squares  6 x 6, MC = 111, Permutations lines n and (7 - n):

Attachment 6.5.4

Magic Squares  6 x 6, MC = 111, Permutations lines 1, 2, 3 and 4, 5, 6

Attachment 6.5.5

Magic Squares  6 x 6, MC = 111, Combined Permutations

Attachment 6.6.2

Magic Squares  6 x 6, MC = 111, Symmetrical Diagonals (a)

Attachment 6.6.3

Magic Squares  6 x 6, MC = 111, Symmetrical Diagonals (b)

Attachment 6.7.1

Magic Squares  6 x 6, MC = 111, Class {Aijk}

Symmetrical Diagonals

Attachment 6.8.1

Medjig Squares 3 x 3, MC = 9

Attachment 6.8.2

Magic Squares  6 x 6, MC = 111, Deducted from Medjig Squares

Attachment 6.8.3

Medjig Squares 3 x 3, MC = 9  , Symmetrical Main Diagonals

Attachment 6.8.4

Magic Squares  6 x 6, MC = 111, Deducted from Medjig Squares

Attachment 6.9.1

Magic Squares  6 x 6, MC = 111, Concentric Squares (1)

Attachment 6.9.2

Magic Squares  6 x 6, MC = 111, Concentric Squares (2)

Attachment 6.9.5

Magic Squares  6 x 6, MC = 111, Concentric Squares (3)

Attachment 6.9.4

Magic Squares  6 x 6, MC = 111, Eccentric  Squares

Attachment 6.7.4

Magic Squares  6 x 6, MC = 111, Inlaid Magic Squares

Attachment 6.10.2

Magic Squares  6 x 6, MC = 111, Partly Rectangular Compact

Attachment 6.10.3

Magic Squares  6 x 6, MC = 111, Rectangular Compact

Attachment 6.10.4

Magic Squares  6 x 6, MC = 111, Rectangular Compact, Axial Symmetric

Attachment 6.11.2

Magic Squares  6 x 6, MC = 111, Axial Symmetric

Attachment 6.11.3

Magic Squares  6 x 6, MC = 111, Row Symmetric (Type 1)

Attachment 6.11.4

Magic Squares  6 x 6, MC = 111, Row Symmetric (Type 2)

Attachment 6.11.6

Magic Squares  6 x 6, MC = 111, Symmetric (Type 3)

Attachment 6.11.7

Magic Squares  6 x 6, MC = 111, Symmetric (Type 4)

Attachment 6.11.8

Magic Squares  6 x 6, MC = 111, Symmetric (Type 5)

Attachment 6.11.9

Magic Squares  6 x 6, MC = 111, Symmetric (Type 6)

Attachment 6.2.2

Magic Squares  6 x 6, MC = 111, Bimagic Diagonals

Attachment 6.2.4

Magic Squares  6 x 6, MC = 111, Bimagic Columns


Magic Squares (6 x 6)
Non Consecutive Integers:

Attachment 6.10.2

Pan Magic Squares  6 x 6, MC = 150, Most Perfect

Attachment 6.10.3

Pan Magic Squares  6 x 6, MC = 150, 2 x 2 Compact, Class Example

Attachment 6.10.4

Pan Magic Squares  6 x 6, MC = 150, Associated

Attachment 6.10.5

Pan Magic Squares  6 x 6, MC = 120, Concentric, Associated Center Squares

Attachment 6.10.7

Pan Magic Squares  6 x 6, MC = 150, Compact Ultra Magic

Attachment 6.10.8a

Pan Magic Squares  6 x 6, MC = 120, Pan Magic, Complete, Semi Mgc Sub Sqrs

Attachment 6.10.8b

Pan Magic Squares  6 x 6, MC = 120, Associated, Semi Mgc Sub Sqrs

Attachment 6.10.5

Magic Squares  6 x 6, MC = 120, Rectangular Compact, Row Symmetric

Attachment 6.11.5

Magic Squares  6 x 6, MC = 120, Crosswise Symmetric

Attachment 6.12.2

Pan Magic Squares  6 x 6, MC = 120, Axial Symmetric

Attachment 6.12.3

Pan Magic Squares  6 x 6, MC = 126, Row Symmetric (Type 1)

Attachment 6.12.4

Pan Magic Squares  6 x 6, MC = 120, Row Symmetric (Type 2)

Attachment 6.12.5

Pan Magic Squares  6 x 6, MC = 120, Crosswise Symmetric

Attachment 6.12.7

Pan Magic Squares  6 x 6, MC = 120, Symmetric (Type 4)

Attachment 6.13.24

Partly Bimagic Squares  6 x 6, MC = 150, Bimagic Rows and Columns

Attachment 6.13.34

Partly Bimagic Squares  6 x 6, MC = 168, One Bimagic Diagonal

Attachment 6.13.44

Bimagic Squares  6 x 6, MC = 408, Associated

Attachment 6.13.54

Bimagic Squares  6 x 6, MC = 219, Crosswise Symmetric

Attachment 6.13.64

Bimagic Squares  6 x 6, MC = 330, Crosswise Symmetric

Magic Squares (7 x 7):

Attachment 7.1.1

Pan Magic Squares   7 x 7, MC = 175, b(i) - 1 =< a(i) =< b(i) + 1

Attachment 7.3

Pan Magic Squares   7 x 7, MC = 175, Collection {Aijk}

Attachment 7.3.4

Sudoku Squares      7 x 7, MC =  21, Pan Magic

Attachment 7.4.1

Ultra Magic Squares 7 x 7, MC = 175, a(49) = 1, a(48) = 9, a(47) = 27

Attachment 7.4.2

Ultra Magic Squares 7 x 7, MC = 175, Three Cell Patterns

Attachment 7.4.3

Sudoku Squares      7 x 7, MC =  21, Ultra Magic

Attachment 7.4.4

Ultra Magic Squares 7 x 7, MC = 175, Based on Attachment 7.4.3

Attachment 7.5.1

Magic Squares       7 x 7, MC = 175, Concentric Squares (1)

Attachment 7.5.2

Magic Squares       7 x 7, MC = 175, Concentric Squares (2)

Attachment 7.5.3

Magic Squares       7 x 7, MC = 175, Eccentric  Squares (1)

Attachment 7.5.4

Magic Squares       7 x 7, MC = 175, Eccentric  Squares (2)

Attachment 7.6.1

Magic Squares       7 x 7, MC = 175, Overlapping Sub Squares (1)

Attachment 7.6.2

Magic Squares       7 x 7, MC = 175, Overlapping Sub Squares (2)

Attachment 7.6.3

Magic Squares       7 x 7, MC = 175, Overlapping Sub Squares (3)

Attachment 7.6.4

Magic Squares       7 x 7, MC = 175, Overlapping Sub Squares (4)

Attachment 7.6.5

Pan Magic Squares   5 x 5, MC = 125, Corner = 25

Attachment 7.6.6

Magic Squares       7 x 7, MC = 175, Overlapping Sub Squares (6)

Attachment 7.6.7

Ass Magic Squares   7 x 7, MC = 175, Overlapping Sub Squares (7)

Attachment 7.6.8

Ass Magic Squares   7 x 7, MC = 175, Overlapping Sub Squares (8)

Attachment 7.8.7

Comp. Magic Squares 7 x 7, MC = 175, Ass Sub Squares (1)

Attachment 7.8.8

Comp. Magic Squares 7 x 7, MC = 175, Ass Sub Squares (2)

Attachment 7.7.1

Bordered Mgc Sqrs   7 x 7, MC = 175, Square and Diamond Inlay

Attachment 7.7.2

Ass Magic Squares   7 x 7, MC = 175, Diamond Inlays Order 3 and 4

Attachment 7.7.3

Ass Magic Squares   7 x 7, MC = 175, Square  Inlays Order 3 and 4

Attachment 7.7.4

Ultra Magic Squares 7 x 7, MC = 175, Conc Square and Square Inlay (1)

Attachment 7.7.5

Ultra Magic Squares 7 x 7, MC = 175, Conc Square and Square Inlay (2)

Attachment 7.7.6

Ultra Magic Squares 7 x 7, MC = 175, Square Inlays

Attachment 7.7.7

Ultra Magic Squares 7 x 7, MC = 175, Conc Square and Diamond Inlay

Attachment 7.7.8

Ultra Magic Squares 7 x 7, MC = 175, Square and Diamond Inlay (1)

Attachment 7.7.9

Ultra Magic Squares 7 x 7, MC = 175, Square and Diamond Inlay (2)

Attachment 7.8.1

Magic Squares       7 x 7, MC = 175, Bent Diagonals

Attachment 7.8.2

Magic Squares       7 x 7, MC = 175, Bent Diagonals, Symm Axes and Dia

Attachment 7.8.3

Pan Magic Squares   7 x 7, MC = 175, Bent Diagonals

Attachment 7.8.4

Pan Magic Squares   7 x 7, MC = 175, Bent Diagonals, Symm Axes and Dia

Attachment 7.8.5

Magic Squares       7 x 7, MC = 175, Bent Diagonals, Properties

Magic Squares (8 x 8):

Attachment 8.1.3

Pan Magic Squares 8 x 8, MC = 260, Class {Aijk}

Attachment 8.1.4

Pan Magic Squares 8 x 8, MC = 260, Collection {B}

Attachment 8.1.5

Pan Magic Squares 8 x 8, MC = 260, Collection {Aijk} based on Square 1

Attachment 8.1.6

Pan Magic Squares 8 x 8, MC = 260, Unique Base Squares

Attachment 8.2.1

Pan Magic Squares 8 x 8, Composed of Pan Magic Sub Squares (4 x 4)

Attachment 8.2.2

Magic Squares     8 x 8, Composed of Pan Magic Sub Squares (4 x 4)

Attachment 8.2.3

Magic Squares     8 x 8, Composed of Magic Sub Squares (4 x 4)

Attachment 8.2.4

Pan Magic Squares 8 x 8, Composed of Pan Magic Sub Squares (4 x 4)
Containing Magic Middle Squares (4 x 4)

Attachment 8.2.5

Magic Squares     8 x 8, Composed of Magic Sub Squares (4 x 4)
Containing Magic Middle Squares (4 x 4), Bent Diagonals

Attachment 8.2.6

Magic Squares     8 x 8, Composed of Associated Magic Sub Squares (4 x 4)

Attachment 8.2.7

Magic Squares     8 x 8, Composed of Associated Magic Sub Squares (4 x 4)
Miscellaneous Valid Variable Ranges

Attachment 8.3.1

Medjig Squares    4 x 4, MC =  12
Main Diagonals, Distinct Integers

Attachment 8.3.2

Magic Squares     8 x 8, MC = 260, Deducted from Medjig Squares
Main Diagonals, Distinct Integers

Attachment 8.3.3

Magic Squares     8 x 8, MC = 260, Deducted from Medjig Squares
Pan Magic Complete Squares

Attachment 8.3.4

Magic Squares     8 x 8, MC = 260, Deducted from Medjig Squares
Associated Squares

Attachment 8.4.1

Franklin Squares  8 x 8, MC = 260, Main and Parallel Bent Diagonals

Attachment 8.4.2

Franklin Squares  8 x 8, MC = 260, a(64) = 17 and a(63) = 32

Attachment 8.4.3

Franklin Squares  8 x 8, MC = 260, More Patterns summing to 260

Attachment 8.4.4

Franklin Squares  8 x 8, MC = 260, a(64) = 17 and a(63) = 40
Barink Restrictions

Attachment 8.5.1

Pan Magic Squares 8 x 8, Franklin Properties 2 and 5

Attachment 8.5.2

Pan Magic Squares 8 x 8, Composed of Pan Magic Sub Squares (4 x 4)
Franklin Property 5

Attachment 8.5.4

Pan Magic Squares 8 x 8, Composed of Pan Magic Sub Squares (4 x 4)
Franklin Properties 2 and 5, Barink Restrictions

Attachment 8.5.5

Pan Magic Squares 8 x 8, Most Perfect (Compact and Complete)

Attachment 8.6.1

Magic Squares     8 x 8, MC = 260, Concentric Squares (1)

Attachment 8.6.2

Magic Squares     8 x 8, MC = 260, Concentric Squares (2)

Attachment 8.6.3

Magic Squares     8 x 8, MC = 260, Eccentric Squares

Attachment 8.6.4

Magic Squares     8 x 8, MC = 260, Bordered Squares

Attachment 8.6.5

Ultra Mgc Sqrs    8 x 8, MC = 260, Compact

Attachment 8.6.7

Ultra Mgc Sqrs    8 x 8, MC = 260, Non Overlapping Sub Squares (a)

Attachment 8.6.8

Ultra Mgc Sqrs    8 x 8, MC = 260, Non Overlapping Sub Squares (b)

Attachment 8.6.9

Pan Magic Squares 8 x 8, MC = 260, Non Overl Sub Squares, Complete

Attachment 8.6.10

Ultra Mgc Sqrs    8 x 8, MC = 260, Rectangular Compact

Attachment 8.6.11

Pan Magic Squares 8 x 8, MC = 260, Rectangular Compact, Complete

Attachment 8.6.12

Magic Squares     8 x 8, MC = 260, Parttly Rect Compact, Associated

Attachment 8.6.13

Magic Squares     8 x 8, MC = 260, Parttly Rect Compact, Complete

Attachment 8.6.14

Magic Squares     8 x 8, MC = 260, Unique Inlaid Magic Squares

Attachment 8.7.1

Magic Squares     8 x 8, MC = 260, Composed of Magic Sub Squares

Attachment 8.7.2

Pan Magic Squares 4 x 4, MC = 130

Attachment 8.7.3

Sudoku Squares    8 x 8, MC =  28

Attachment 8.7.5

Quaternary Sqrs   Examples

Attachment 8.7.51

Quaternary Sqrs   8 x 8, MC =  12, Pan Magic, Associated

Attachment 8.7.52

Quaternary Sqrs   8 x 8, MC =  12, Pan Magic, Complete

Magic Squares (9 x 9):

Attachment 9.3

Pan Magic Squares 9 x 9, MC = 369, Class {Aijk}

Attachment 9.3.1

Pan Magic Squares 9 x 9, MC = 369, Collection {B}

Attachment 9.3.2

Pan Magic Squares 9 x 9, MC = 369, Collection {Aijk} based on Square 1

Attachment 9.3.3

Pan Magic Squares 9 x 9, MC = 369, Unique Base Squares

Attachment 9.4.1

Compact Associated Pan Magic Squares, MC = 369

Attachment 9.4.2

Compact Pan Magic Squares MC = 369
Every third-row and third-column summing to 123

Attachment 9.4.3

Compact Associated Pan Magic Squares MC = 369
Every third-row and third-column summing to 123

Attachment 9.4.4

Compact Associated Pan Magic Squares MC = 369, a(81) = 1
Every third-row and third-column summing to 123

Attachment 9.5.1

Magic Squares     9 x 9, MC = 369, Concentric Squares (1)

Attachment 9.5.2

Magic Squares     9 x 9, MC = 369, Concentric Squares (2)

Attachment 9.5.8

Magic Squares     9 x 9, MC = 369, Bordered Squares

Attachment 9.5.3

Magic Squares     9 x 9, Eccentric Squares, Eccentric Corner Squares

Attachment 9.5.4

Magic Squares     9 x 9, Eccentric Squares, s1 = 369, s2 = 209

Attachment 9.5.5

Base Squares for  9 x 9, Eccentric Squares, s1 = 369, s2 = 209

Attachment 9.5.6

Magic Squares     9 x 9, Eccentric Squares, Overlapping Sub Squares

Attachment 9.5.7

Pan Magic Squares 9 x 9, Compact, Associated

Attachment 9.6.1

Sudoku Squares    9 x 9, MC = 36, Associated Compact Pan Magic Squares

Attachment 9.6.2

Ternary Squares   9 x 9, MC =  9, John Hendricks

Attachment 9.6.3

Sudoku Squares    9 x 9, MC = 36, John Hendricks

Attachment 9.6.4

Sudoku Squares    9 x 9, MC = 36, Non-Associated Compact Pan Magic Squares

Attachment 9.6.5

Ternary Squares   Examples

Attachment 9.6.6

Ternary Squares   9 x 9, MC =  9, Non-Associated Partly Compact

Attachment 9.6.7

Sudoku Squares    9 x 9, MC = 36, Non-Associated Partly Compact

Magic Squares (10 x 10):

Attachment 10.1.1

Medjig Squares     5 x  5, MC =   15

Attachment 10.1.2

Magic Squares     10 x 10, MC =  505, Deducted from Medjig Squares

Attachment 10.1.3

Conc Mgc Sqrs     10 x 10, MC =  505, Deducted from Medjig Squares

Attachment 10.1.4

Ecc  Mgc Sqrs     10 x 10, MC =  505, Deducted from Medjig Squares

Attachment 10.1.7

Magic Squares     10 x 10, MC =  505, Bimagic Diagonals

Attachment 10.1.9

Magic Squares     10 x 10, MC =  505, Bimagic Center Lines

Attachment 10.2.1

Magic Squares     10 x 10, MC =  505, Concentric Squares (1)

Attachment 10.2.2

Magic Squares     10 x 10, MC =  505, Concentric Squares (2)

Attachment 10.2.6

Magic Squares     10 x 10, MC =  505, Bordered Squares

Attachment 10.2.3

Magic Squares     10 x 10, Eccentric Squares, s1 = 505, s2 = 199

Attachment 10.2.4

Magic Squares     10 x 10, Eccentric Squares, s1 = 505, s2 = 312

Attachment 10.2.5a

Composed Squares  10 x 10, MC =  505, Conc Mgc Crnr Sqr

Attachment 10.2.5b

Composed Squares  10 x 10, MC =  505, Ecc  Mgc Crnr Sqr

Attachment 10.2.6a

Composed Squares  10 x 10, MC =  505, Type 1, Conc Mgc Crnr Sqr

Attachment 10.2.6b

Composed Squares  10 x 10, MC =  505, Type 1, Ecc  Mgc Crnr Sqr

Attachment 10.2.7a

Composed Squares  10 x 10, MC =  505, Type 2, Conc Mgc Crnr Sqr

Attachment 10.2.7b

Composed Squares  10 x 10, MC =  505, Type 2, Ecc  Mgc Crnr Sqr

Attachment 10.2.8

Composed Squares  10 x 10, MC =  505, Type 3, Ecc  Mgc Crnr Sqr

Attachment 10.2.9

Composed Squares  10 x 10, MC =  505, Type 4, Center Crosses

Attachment 10.3.2

Pan Magic Squares 10 x 10, MC = 2250, Most Perfect

Attachment 10.3.3

Ultra Mgc Sqrs    10 x 10, MC = 3510, Compact

Magic Squares (11 x 11):

Attachment 14.9.1

Magic Squares     11 x 11, MC =  671, Borders

Attachment 14.9.2

Magic Squares     11 x 11, MC =  671, Bordered

Attachment 14.9.4

Magic Squares     11 x 11, MC =  671, Eccentric

Attachment 14.9.6

Magic Squares     11 x 11, MC =  671, Composed Border

Attachment 14.9.7

Magic Squares     11 x 11, MC =  671, Associated, Composed (1)

Attachment 14.9.8

Magic Squares     11 x 11, MC =  671, Associated, Composed (2)

Attachment 14.9.9b

Magic Squares     11 x 11, MC =  671, Associated, Inlaid (1)

Attachment 14.8.16

Magic Squares     11 x 11, MC =  671, Associated, Inlaid (2)

Attachment 14.8.15

Magic Squares     11 x 11, MC =  671, Composed

Attachment 14.8.5.7

Magic Squares     11 x 11, MC =  671, Composed, Overl Sub Sqrs (1)

Attachment 14.9.8.1

Magic Squares     11 x 11, MC =  671, Composed, Overl Sub Sqrs (2)

Attachment 14.9.8.2

Magic Squares     15 x 15, MC = 1695, Composed, Overl Sub Sqrs

Magic Squares (Higher Order):

Attachment 12.1

Barink Squares    12 x 12, MC =  870, a(144) = 72 and a(143) = 74

Attachment 12.2a

Morris Squares    12 x 12, MC =  870,
a(136), a(138) ... a(140) and a(142) ... a(144) constant

Attachment 12.2b

Morris Squares    12 x 12, MC =  870, Most Perfect
a(139) ... a(144) constant

Attachment 12.3

Franklin Squares  16 x 16, MC = 2056 (1)

Attachment 12.4

Franklin Squares  16 x 16, MC = 2056 (2)
Most Perfect Pan Magic Squares

Attachment 12.5

Franklin Squares  16 x 16, MC = 2056 (3)
Most Perfect Pan Magic Squares, Barink Restrictions

Interactive Solutions

Form 4

Pan Magic Squares   4 x  4, MC =   34, Binary Method

Form 5

Pan Magic Squares   5 x  5, MC =   65, Sudoku Method

Form 6

Magic Squares       6 x  6, MC =  111, Medjig Method

Form 7

Pan Magic Squares   7 x  7, MC =  175, Sudoku Method

Form 8

Most Perfect Msqrs  8 x  8, MC =  260, Binary Method

Form 9

Bimagic Squares     9 x  9, MC =  369, Ternary Method

Form 12

Most Perfect Msqrs 12 x 12, MC =  870, Basic Pattern Method

Form 16

Most Perfect Msqrs 16 x 16, MC = 2056, Basic Pattern Method


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