GENERATING MAGIC SQUARES IN LEARNING MATHEMATICS

1

• GENERATING MAGIC SQUARES IN LEARNING MATHEMATICS

Presenter  Dr A.RAMBABU, Professor (M.Sc.(Maths
), M.A.(Eco.), M.Ed., M.Phil.(Maths),
NET,SLETPh.D. in Education) LB College of
Education, Warangal Rtd. Associate Professor in
Govt. College of Teacher Education - Warangal
Telangana - 506009
2
Points to be discuss

• Introduction
• Concept of Magic Square
• Ramanujan Birth day magic Square and its
  application
• Generating Processes of order 3x3 Magic Squares
  including any odd order (5x5, 7x7, . ,) Magic
  Square
• Generating Processes of order 4x4, 8x8, 12x12,
  16x16, . , 4nx4n, n belongs to N
• Other interesting Magic squares
• Inlaid Magic Squares Multiplicative
• magic squares

3
INTRODUCTION

• The magic squares include numbers, figures,
  shapes, patterns, symbols, feelings, thoughts
  etc. which are bases of Mathematics.
• Most of the students learning Mathematics without
  knowing the process / procedure / pattern /
  rhythm of numbers, figures, shapes, patterns,
  symbols, feelings, thoughts etc. They are
  learning only product or result without knowing
  the process and getting highest marks in
  examination even in 21st century.
• Whenever understand the process / procedure /
  pattern / rhythm of numbers, figures, shapes,
  patterns, symbols, feelings, thoughts etc., our
  Mathematics is very easy to apply in different
  situations of daily life.
• The purpose of this presentation is to learn the
  generating process of Magic Squares for creating
  interest, curiosity and self-confidence in
  learning mathematics

4
Magic Square Concept

• A magic square is an arrangement of the numbers
  from 1 to n2 (n-squared) in an order n x n, n3
  matrix, with each number occurring exactly once,
  and such that the sum of the entries of any row,
  any column, or any main diagonal is the same.
• The sum is called as Magic Constant M. It is also
  called as Sum Square Sn or Magic Sum.
• Sn M n(n2 1)/2
• For 3x3, 4x4, 5x5, 6x6, 7x7, 8x8, ., nxn, Magic
  Constants are 15, 34, 65, 111, 175, 260, , n(n2
  1)/2 respectively.

5
Observe the Magic Square Observe the Magic Square Observe the Magic Square Observe the Magic Square Observe the Magic Square 139
Date of Birth of Srinivasa Ramanuan. Magic Square Magic Constant (M) 22121887 139.   22 12 18 87 139
Date of Birth of Srinivasa Ramanuan. Magic Square Magic Constant (M) 22121887 139.   88 17 9 25 139
Date of Birth of Srinivasa Ramanuan. Magic Square Magic Constant (M) 22121887 139.   10 24 89 16 139
Date of Birth of Srinivasa Ramanuan. Magic Square Magic Constant (M) 22121887 139.   19 86 23 11 139
Date of Birth of Srinivasa Ramanuan. Magic Square Magic Constant (M) 22121887 139.   139 139 139 139 139
6
Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139 Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139 Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139 Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139 Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139 Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139 Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139 Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139 Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD 22121887 139     139
A B C D   22 12 18 87 139
D1 C-1 B-3 A3   88 17 9 25 139
B-2 A2 D2 C-2   10 24 89 16 139
C1 D-1 A1 B-1   19 86 23 11 139
  139 139 139 139 139
7
Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square 
A B C D   A 0 B 0 C 0 D 0
D1 C-1 B-3 A3   D 1 C -1 B -3 A 3
B-2 A2 D2 C-2   B -2 A 2 D 2 C -2
C1 D-1 A1 B-1   C 1 D -1 A 1 B -1
         
8
Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD Based on Srinivasa Ramanuan Birth day Magic Square, it is generalized as A Date of Birth, B Month of Birth, C Century of the year of birth, D Birth year two digits Then M ABCD Dr Rambabu Akinapallys Birth Day (as per certificate) is 12-06-1958 and M1206195895. Now, it is perfect magic square. Dr Rambabu Akinapallys Birth Day (as per certificate) is 12-06-1958 and M1206195895. Now, it is perfect magic square. Dr Rambabu Akinapallys Birth Day (as per certificate) is 12-06-1958 and M1206195895. Now, it is perfect magic square. Dr Rambabu Akinapallys Birth Day (as per certificate) is 12-06-1958 and M1206195895. Now, it is perfect magic square.     95
A B C D   12 12 06 19 58 95
D1 C-1 B-3 A3   59 59 18 03 15 95
B-2 A2 D2 C-2   04 04 14 60 17 95
C1 D-1 A1 B-1   20 20 57 13 05 95
  95 95 95 95 95 95
9
In some cases zero, negative numbers and repeated
numbers may also come in the cells which is not
concerned the rules of magic square. Most of the
cases possible to modify. Observe the following
example
10
But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 But, Dr Rambabu Akinapallys actual Birth Day is 18-12-1958. It is not perfect magic square due to 18, 19 20 are repeated twice. But, M107 107
A B C D   18 1 2 19 58 107
D1 C-1 B-3 A3   59 18 09 21 107
B-2 A2 D2 C-2   10 20 60 17 107
C1 D-1 A1 B-1   20 57 19 11 107
  107 107 107 107 107
11
Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square  Srinivasa Ramanujan Birth Day generalized Magic Square 
A B C D   A 0 B 0 C 0 D 0
D1 C-1 B-3 A3   D 1 C -1 B -3 A 3
B-2 A2 D2 C-2   B -2 A 2 D 2 C -2
C1 D-1 A1 B-1   C 1 D -1 A 1 B -1
         
12
THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3 THREE 1S, THREE -1S, TWO 2S, TWO -2S, ONE 3S, ONE -3S Instead of a set 1,23, substitute any 3 numbers of AP (Arithmetic Progression) such as (2,4,6), (3,6,9), , (10,16,22), , (100,200,300), Here substituted 2,4,6 instead of 1,2,3
A 0 B 0 C 0 D 0 A 0 B 0 C 0 D 0 A B C D
D 1 C -1 B -3 A 3 D 2 C -2 B -6 A 6 D2 C-2 B-6 A6
B -2 A 2 D 2 C -2 B -4 A 4 D 4 C -4 B-4 A4 D4 C-4
C 1 D -1 A 1 B -1 C 2 D -2 A 2 B -2 C2 D-2 A2 B-2

13

107
A B C D A 18 B 12 C 19 D 58 18 12 19 58 107
D2 C-2 B-6 A6 D 60 C 17 B 6 A 24 60 17 6 24 107
B-4 A4 D4 C-4 B 8 A 22 D 62 C 15 8 22 62 15 107
C2 D-2 A2 B-2 C 21 D 56 A 20 B 10 21 56 20 10 107
107 107 107 107 107
14
Srinivasa Ramanujan Birth Day generalized Magic Square Srinivasa Ramanujan Birth Day generalized Magic Square Srinivasa Ramanujan Birth Day generalized Magic Square Srinivasa Ramanujan Birth Day generalized Magic Square   Modified generalized Magic Square Modified generalized Magic Square Modified generalized Magic Square Modified generalized Magic Square
A B C D   A B C D
D1 C-1 B-3 A3   D-2 C2 B2 A-2
B-2 A2 D2 C-2   B1 A-3 D-1 C3
C1 D-1 A1 B-1   C1 D1 A-1 B-1
         
15
Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square Generated another Magic Square for the same Date of Birth 18-12-1958, M107 comparing with Ramanujan Birth day magic square 107
A B C D   A B C D 18 12 19 58 107
D1 C-1 B-3 A3   D-2 C2 B2 A-2 56 21 14 16 107
B-2 A2 D2 C-2   B1 A-3 D-1 C3 13 15 57 22 107
C1 D-1 A1 B-1   C1 D1 A-1 B-1 20 59 17 11 107
          107 107 107 107 107
16
Modified Magic Square Modified Magic Square Modified Magic Square Modified Magic Square 107 Based on Srinivasa Ramanujan Birth Day (Set 2,4,6) Based on Srinivasa Ramanujan Birth Day (Set 2,4,6) Based on Srinivasa Ramanujan Birth Day (Set 2,4,6) Based on Srinivasa Ramanujan Birth Day (Set 2,4,6) 107
18 12 19 58 107 18 12 19 58 107
56 21 14 16 107 60 17 6 24 107
13 15 57 22 107 8 22 62 15 107
20 59 17 11 107 21 56 20 10 107
107 107 107 107 107 107 107 107 107 107
17
Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square Generating order 4x4 other magic squares based on Srinivasa Ramanujan Birth Day Magic Square
Ramanujan Birth Day Magic Square 4 Corners 2x2 squares (Quadruples) Ramanujan Birth Day Magic Square 4 Corners 2x2 squares (Quadruples) Ramanujan Birth Day Magic Square 4 Corners 2x2 squares (Quadruples) Ramanujan Birth Day Magic Square 4 Corners 2x2 squares (Quadruples) Quadruples or Every corner 2x2 square convert as a row / column becomes another magic square Quadruples or Every corner 2x2 square convert as a row / column becomes another magic square Quadruples or Every corner 2x2 square convert as a row / column becomes another magic square Quadruples or Every corner 2x2 square convert as a row / column becomes another magic square
A B C D A B D1 C-1
D1 C-1 B-3 A3 C D B-3 A3
B-2 A2 D2 C-2 B-2 A2 C1 D-1
C1 D-1 A1 B-1 D2 C-2 A1 B-1
18
1st type Common usage method
. Write 1 to 9 in 3 rows as 1st figure 2. Put
straight lines in cross as shown in 2nd figure
3. Here, we can get 1,3,7,9 numbers are outside
the 3x3 box and 2,4,6,8 are inside. Shift the
outside numbers corresponding opposite cells of
the 3x3 box as shown in 3rd figure. 4. Then, we
can get required order 3x3 magic square in 3x3
box as shown in 4th figure.
1 2 3 4 4
1 2 3 4
19
1. Middle number 5 place in the middle cell. 2.
The adjacent numbers of 5 (i.e. 46) are place in
the corner cells in opposite sides. It can be 4
types as rotating right or left. Observe, the 2
to 4 squares are the right rotation of 1st
square.3. The last number 9 can be placed in the
cell of the adjacent of 4 (to do not touch /
attach the number 6). 4. The cells are two in
every square. Observe 1a and 1b 9 can be placed
in any one of the two cells. 5. Fill the
remaining numbers. It is now very easy. 6. Above
each square 1a and 1b can be rotated right or
left. We can get each square as 4 squares. So,
total 8 squares.
2nd type
20
(No Transcript)
21
Another METHOD Any odd order Magic Square based
on order 3x3 magic squareStart with 1 in the
middle cell of 1st row. Rule 1 Move the arrow
mark always through the right corner to upper row
(Example i) or move the arrow mark always through
the left corner to upper row (Example ii). In
this i) example followed right movement.Rule 2
If there is no cell for the arrow, come to the
bottom of the row or column put the number
outside and follow the arrow marks as like 1 2.
Rule 3 If arrow mark resisted its movement by
any number, moves the arrow mark to the below
cell.Rule 4 If arrow mark reaches through the
corner of the square, moves the arrow mark to the
below cell as like 6.
Note 1. Rotation of i) or ii) right side or left
side each magic square gets 4 types of magic
squares. Total 3x3 magic squares are 8 types as
previously explained. Note2. This 3x3 generating
model applicable to generate all odd magic
squares i.e 3x3, 5x5, 7x7, 9x9, , 1001x1001,
See the 5x5 magic square a model generation.
22
The 5x5 magic square a model generation














23
Generating 4x4 magic squaresThis way of
constructing a 4x4 magic square can be found in.
Draw a 4x4 square and write 1 to 16 numbers.
Draw lines for the two diagonals. Observe the
red numbers which are fixed in their concerned
cells. Remove remaining numbers from their cells.
Then count down from 16 to 1, and using only the
numbers not yet in the diagonals of the square,
fill in the boxes that are left other than
diagonal. (see numbers in 2nd square below).
Think other than this type of generating magic
squares.
24
Order 8x8 normal Magic Square
 
25
(No Transcript)
26
Order 16x16 normal Magic Square
27
Further Suggestions

• Further need to identify the generalization
  process of Magic Squares with order 6x6, 10x10,
  14x14, .

28
INLAID MAGIC SQUARES Created by John R.
Hendrick'sSquare 3x3, M S3 123, Square 5x5,
M S5 205, Square 7x7, S7 287,
Square 9x9, M S9 369
29
Magic sums are U.L. 1477, 1055, 633 -- U.R.
1337 -- L.L. 1470, 1050  -- L. R. 1330, 950,
570
30
Four plus five equals nine Magic Square Created
by Kenneth Kelsey of Great Britain
31
Multiplicative magic squares M constant
product of numbers.

Order 3x3, M 216 Order 3x3, M 216 Order 3x3, M 216
2 9 12
36 6 1
3 4 18
Order 4x4, M 6720 Order 4x4, M 6720 Order 4x4, M 6720 Order 4x4, M 6720
1 6 20 56
40 28 2 3
14 5 24 4
12 8 7 10
Order 3x3, M 32768 Order 3x3, M 32768 Order 3x3, M 32768
16 512 4
8 32 128
256 2 64
32
Skallis multiplicative 7 x 7, M6,227,020,800 Skallis multiplicative 7 x 7, M6,227,020,800 Skallis multiplicative 7 x 7, M6,227,020,800 Skallis multiplicative 7 x 7, M6,227,020,800 Skallis multiplicative 7 x 7, M6,227,020,800 Skallis multiplicative 7 x 7, M6,227,020,800 Skallis multiplicative 7 x 7, M6,227,020,800
27 50 66 84 13 2 32
24 52 3 40 54 70 11
56 9 20 44 36 65 6
55 72 91 1 16 36 30
4 24 45 60 77 12 26
10 22 48 39 5 48 63
78 7 8 18 40 33 60
33
Skalli multiplicative 7 x 7 of complex numbers, M-352,507,340,640 - 400,599,719,520 i Skalli multiplicative 7 x 7 of complex numbers, M-352,507,340,640 - 400,599,719,520 i Skalli multiplicative 7 x 7 of complex numbers, M-352,507,340,640 - 400,599,719,520 i Skalli multiplicative 7 x 7 of complex numbers, M-352,507,340,640 - 400,599,719,520 i Skalli multiplicative 7 x 7 of complex numbers, M-352,507,340,640 - 400,599,719,520 i Skalli multiplicative 7 x 7 of complex numbers, M-352,507,340,640 - 400,599,719,520 i Skalli multiplicative 7 x 7 of complex numbers, M-352,507,340,640 - 400,599,719,520 i
2114i -7030i -93-9i -105-217i 1650i 4-14i 14-8i
63-35i 28114i -14i 26i 3-11i 211357i -123-87i
31-15i 13-13i -10369i -261-213i 49-49i -462i -62i
102-84i -28-14i 43247i -10-2i 59i 31-27i -7791i
-22-6i 77i 814i 5020i -525-492i -28-42i -7317i
5468i 138-165i -56-98i -6335i 4-8i 2-4i 70-53i
2422i -46-16i 6-4i 1720i 110160i 84-189i 42-14i
34
THANK YOU ONE AND ALL
35
3 x 3 Magic Square By Srinivasa Ramanujan A,B,C
P,Q,R are any two sets in AP (Arithmetic
Progression)
A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 A,B,C5,8,11 PQR10,15,20 69
CQ AP BR 1115 510 820 26 15 28 69
AR BQ CP 520 815 1110 25 23 21 69
BP CR AQ 810 1120 515 18 31 20 69
69 69 69 69
36
111
1 32 33 34 6 5 111
25 11 28 27 8 12 111
18 23 16 15 19 20 111
24 17 21 22 13 14 111
7 26 10 9 30 29 111
36 2 3 4 35 31 111
111 111 111 111 111 111 111
 
37
THANKS