Monday, September 2, 2013

72-Magic Square - Different cell elements and addition

Addition of different cell elements of Magic Squares:

We have seen the addition of the elements in rows columns, and diagonals are the same in all the cases. Each row or column or diagonal contains 4 elements. Now we see the elements in different cases or places will also show the addition as 34.

The following diagram gives us the details of all possible 4 places or cells which we are considering as our elements.

(1 + 15 + 12 + 6)      (14 + 4 + 7 + 9)
(R1C1, R1C2, R2C1, R2C2)    (R1C3, R1C4, R2C3, R2C4)

Here Sum of all these numbers is 34 (1 + 15 + 12 + 6). In the same way, we can observe the same sum for all other diagrams.
Now we will see the remaining groups of different places of cells taken in to account to get the sum as 34

(11 + 5 + 2 + 16)      (8 + 10 + 13 + 3)
(R3C3, R3C4, R4C3, R4C4)  (R3C1, R3C2, R4C1, R4C2)

(15 + 14 + 3 + 2)      (8 + 12 + 9 + 5)
(R1C2, R1C3, R4C2, R4C3)  (R2C1, R3C1, R2C4, R3C4)

(15 + 12 + 5 + 2)      (14 + 9 + 8 + 3)
(R1C2, R2C1, R3C4, R4C3)  (R1C3, R2C4, R3C1, R4C2)

(6 + 7 + 10 + 11)         (1 + 4 + 13 + 16)
(R2C2, R2C3, R3C2, R3C3)  (R1C1, R1C4, R4C1, R4C4)

(1 + 7 + 10 + 16)         (4 + 6 + 11 + 13)
(R1C1, R2C3, R3C2, R4C4)  (R1C4, R2C2, R3C3, R4C1)

(6 + 9 + 3 + 16)         (15 + 4 + 10 + 5)
(R2C2, R2C4, R4C2, R4C4)  (R1C2, R1C4, R3C2, R3C3)

(12 + 7 + 13 + 2)         (1 + 14 + 8 + 11)
(R2C1, R2C3, R4C1, R4C3)  (R1C1, R1C3, R3C1, R3C3)
Like this, you can also find so many relations between the numbers reflected in cells and their addition.

Anil Satpute