শুক্রবার, ৩ জুন, ২০১১

Table of Zero balance



Numerical Zero Balance: Numerical Zero Balance means some numbers combination in 3x3 or 4x4 or 5x5 square shape which always would follow some specific properties.

Firstly we work with 3x3 numerical zero-balance table like below for understand:

Table of Zero balance:

a11
a12
a13
a21
a22
a23
a31
a32
a33


Here the below equations are satisfied :

1. a11 +a12+ a13= 0     4. a11 +a21+ a31= 0     7. a11 +a22+ a33= 0
2. a21 +a22+ a23= 0     5. a12 +a22+ a32= 0     8. a13 +a22+ a31= 0
3. a31 +a32+ a33= 0     6. a13 +a23+ a33= 0
9. a11 + a12+ a13 + a21 + a22 + a23 + a31 + a32+ a33 = 0        
10. a11 + a33= 0          11. a13 + a31= 0
12. a21 + a23= 0          13. a12 + a32= 0           14. a22+ a33= 0



One can get easily find out the values are

1
-4
3
2
0
-2
-3
4
-1


& Finally
1+n
-4-n
3+n
2+n
0
-2-n
-3-n
4+n
-1-n

n= 1,2,3,4,………….∞

So, There are infinite numbers of 3x3 shape numerical zero-balance data.
But the wonder of specialities belongs to 4x4 shape numerical zero-balance data.

Table of Zero balance:

a11
a12
a13
a14
a21
a22
a23
a24
a31
a32
a33
a34
a41
a42
a43
a44

Find out the value of

a11,a12,a13,a14,a21,a22,a23,a24,a31,a32,a33,a34,a41,a42,a43 & a44.

Where a11≠a12≠a13≠a14≠a21≠a22≠a23≠a24≠a31≠a32≠a33≠a34≠a41≠a42≠a43≠a44≠0
and
a11+a12+a21+a22 = 0,                   a12+a13+a22+a23 = 0,             a13+a14+a23+a24 = 0,
a21+a22+a31+a32 = 0,              a22+a23+a32+a33 = 0,                  a23+a24+a33+a34 = 0,
a31+a32+a41+a42 = 0,               a32+a33+a42+a43 = 0,             a33+a34+a43+a44 = 0,
a11+a21+a41+a24 = 0,              a21+a31+a24+a34 = 0,             a31+a41+a34 +a44= 0,
a11+a12+a41+a42 = 0,                   a12+a13+a42+a43 = 0,             a13+a14+a43+a44 = 0,
a11+a21+a14+a24 = 0,                   a21+a31+a24+a34 = 0,             a31+a41+a34+a44 = 0,
a11+a22+a33+a44 = 0,              a14+a23+a32+a41 = 0,             a11+a14+a41+a44 = 0,
a11+a13+a31+a33 = 0,              a12+a14+a32+a34 = 0,             a21+a23+a41+a43 = 0,
a22+a24+a42+a44 = 0,              a12+a21+a34+a43 = 0,             a13+a24+a31+a42 = 0,
a11+a24+a33+a42 = 0,                   a13+a22+a31+a44 = 0,             a14+a21+a32+a43 = 0,
a12+a23+a34+a41 = 0,                     
  a11+a33 = 0,         a22+a44 = 0,       a14+a32 = 0,         a41+a23 = 0,     
 a12+a34 = 0,          a13+a31 = 0,       a21+a43 = 0,         a42+a24 = 0 
Component’s sum of every row and column is zero & above all
a11+a12+a13+a14+a21+a22+a23+a24+a31+a32+a33+a34+a41+a42+a43+a44 = 0

Total 48 zero equations satisfied the table’s numerical combination.

Note: The beginning number may be ±n (where n=1,2,3,............), as one can choose.
 (There are billion billion actually infinite numerical combination fulfills those conditions. Let find out a single one. If one gets any combination then he may find out every combination which fulfills all of those conditions.)
Finaly today 26-5-2015, time 10.00AM, BST, I'm going to reveal the correct answer of above post.
The answer belows here:

1 -3 7 -5
-2 4 -8 6
-7 5 -1 3
8 -6 2 -4

Thanks to all who have tried hard to solve ever.