- 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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.

The 5x5 magic square a model generation

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.

Order 8x8 normal Magic Square

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Order 16x16 normal Magic Square

Further Suggestions

- Further need to identify the generalization

process of Magic Squares with order 6x6, 10x10,

14x14, .

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

Magic sums are U.L. 1477, 1055, 633 -- U.R.

1337 -- L.L. 1470, 1050 -- L. R. 1330, 950,

570

Four plus five equals nine Magic Square Created

by Kenneth Kelsey of Great Britain

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

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

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

THANK YOU ONE AND ALL

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

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

THANKS