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Monday, 23 July 2012

Magic Square Box and others

In this post I'm going to include some pictures which I made my innovation General formula in my school days,

My innovation projects named  " Generalized Starting number for odd numbered Magic Square Boxes " and you can view those pictures here....

I have participated in district level school competitions of Maths and Science Expo at Anderson Matriculation School, Kanchipuram and this is a material I have exhibited in that event on 2004 (on my 11th standard) and won third prize
and before that no one innovated this formula.

What is Magic Square Box?

It's nothing but when some numbers arranged in matrix square format such as 3 row 3 column and 6 row 6 columns, when you gonna add some numbers in row wise or column wise or diagonal wise, you will get some unique result.

Example 

4  9  2
3  5  7
8  1  6

in that above example when add the numbers row wise or column wise or diagonal wise you get the unique result as 15.

But again now I've made some advancement and evolution in this formulas.


in the above picture I'll translate into English

Odd Numbered Magic Square Box 3, 5, 7, 9,11, ........up to 2n-1 (and n is not equal to 0, 1)

" Generalized Starting number formula for Odd numbered Magic Square Box ":

              S(odd)  = [ N(odd) / Odd ] - [ (Odd^2  - 1) /2]

Terms detail :

S(odd) = Starting number of the particular Magic square box number series ( From the above example 1 is starting number of that 3*3 magic square box. so S(odd) = 1 ) and the series of numbers ranges from 1 to 9 ( 1 ,2,3,4,5,6,7,8,9 )

N(odd) = Unique Resultant number when you add up each row or column or diagonal you will get this unique value ( From the above example 15 is the unique value N(odd) =15 )

Odd = It's a type of square matrix form that you gonna be used it ( in the above example I've used 3*3 square matrix format so, Odd = 3  )

Condition:


  • N(Odd) number must be divisible by Odd
  • From the above example 15 is divisible 3 in 3*3 square matrix, other then if you use 7*7 then 714 is also divisible by 7 but 699 cannot be a valid number for 7*7 because it cannot be divisible by 7. 

    This below example shows the 3*3 Magic Square ( square matrix ) format
SEQUENCE
S(odd), S(odd) + 1,  S(odd) + 2, ....... upto S(odd) + (Odd^2-1)   




























I'll show another picture for step by step solution of the problem
in this above picture you can see that shrunk format of the formula

 S(three) = [N(three) / 3 ] - 4
and the below picture shows how that formula can be shrunk from the " Generalized Starting number formula for Odd numbered Magic Square Box "

and finally the condition is that N(three) number must be divisible by 3 ( from the above example N(three) = 15 which has be divisible by 3 and that condition is fully satisfied).

so as per the example 

N(three)  = 15 and Odd = 3
then

S(three) = ( 15 / 3 ) - 4 = 5 - 4 = 1

so S(three)  = 1......... from that 1 is starting number of the sequence
  
we know that sequence

S(odd), S(odd) + 1,  S(odd) + 2, ....... upto S(odd) + (Odd^2-1)   

so in that example sequence

1, 2, 3, 4, 5, 6, 7, 8, 9. 


As same way the below pictures shows 5 * 5 Magic Square box

first one (N(five) = 65, S (five) = 1, and Odd = 5)

and next one shows the shrunk format

in this above picture you can see that shrunk format of the formula

 S(five) = [N(five) / 5 ] - 12

here you got the sequence ranging from 1 to 25

N(five) = 65,
S (five) = 1, 
and Odd = 5 

Same as you can do it for 7*7, 9*9, ..... upto infinite of odd numbers (2n - 1).....

Sample 25 * 25 ........in this you can get 7825 as a value when add numbers from each row or column or diagonal.....




this is all of my innovative formula on Magic Square Box for
" Generalized Starting number formula for Odd numbered Magic Square Box ":

              S(odd)  = [ N(odd) / Odd ] - [ (Odd^2  - 1) /2] 

Condition:


  • N(Odd) number must be divisible by Odd
  • From the above example 15 is divisible 3 in 3*3 square matrix, other then if you use 7*7 then 714 is also divisible by 7 but 699 cannot be a valid number for 7*7 because it cannot be divisible by 7. 

    This below example shows the 3*3 Magic Square ( square matrix ) format
SEQUENCE
S(odd), S(odd) + 1,  S(odd) + 2, ....... upto S(odd) + (Odd^2-1)
  and this now I made one of the advancement and evolution of this formula because the same formula can also be applicable to Even numbered Magic Square Box too but the condition is differed so

S(even)  = [ N(even) / Even ] - [ (even^2  - 1) /2]

and then the

Conditions
  • N(even) must not be divisible by Even
  • But,  N(even) must be divisible at least by Even / 2

    SEQUENCE
    S(even), 
    S(

    even
    ) + 1,  S(even) + 2, ....... upto S(even) + (even^2-1)  

    first one


    second one

    last one



    still I'm doing some advancement over this formula's, and it'ssssssss continue,,,,,,

    See my all pictures of this kind of projects  ( click this below link and view it)

    https://plus.google.com/u/0/photos/115588399550578133989/albums/5766837310431770049