72-Magic Square-5 (different cells and addition part-2)

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Addition of different cell elements of Magic Squares:

🔢 Delving Deeper into Hidden Patterns in the Magic Square  

We have already observed that in a 4 × 4 magic square, the total numbers in every row, column, and diagonal consistently add up to 34.  

That’s the charm of a magic square!  

Each of these lines comprises four elements; this unchanging total is one of its key characteristics.  

However, what’s even more intriguing is that the sum of numbers arranged in particular patterns or positions throughout the square, in addition to rows, columns, or diagonals, also totals 34.  

📐 The illustration below highlights various combinations of 4 cells that adhere to this unique property.  

It’s akin to uncovering concealed symmetry and mathematical harmony that goes beyond the obvious.

 (1 + 15 + 12 + 6)      (14 + 4 + 7 + 9)

(R₁C₁, R₁C₂, R₂C_1, R₂C₂)    (R₁C₃, R₁C₄, R₂C_3, R₂C₄)

🔍 Discovering More Concealed Groups in the Magic Square

In the provided example, the total of the chosen numbers — 1 + 15 + 12 + 6 — equals 34.

This verifies the magical characteristic even when numbers are selected from non-linear positions!

Similarly, numerous other combinations of 4 cells, located in various square areas, also yield the same sum.

✨ Now let's delve into additional groups where the numbers may not align in a straight line, yet their sum still fantastically amounts to 34.

These instances beautifully showcase the hidden symmetry and refined structure of a magic square.

 

(11 + 5 + 2 + 16)      (8 + 10 + 13 + 3)

(R₃C₃, R₃C₄, R₄C_3, R₄C₄)  (R₃C₁, R₃C₂, R₄C_1, R₄C₂)

(15 + 14 + 3 + 2)      (8 + 12 + 9 + 5)

(R₁C₂, R₁C₃, R₄C_2, R₄C₃)  (R₂C₁, R₃C₁, R₂C_4, R₃C₄)

(15 + 12 + 5 + 2)      (14 + 9 + 8 + 3)

(R₁C₂, R₂C₁, R₃C_4, R₄C₃)  (R₁C₃, R₂C₄, R₃C_1, R₄C₂)

(6 + 7 + 10 + 11)         (1 + 4 + 13 + 16)

(R₂C₂, R₂C₃, R₃C_2, R₃C₃)  (R₁C₁, R₁C₄, R₄C_1, R₄C₄)

(1 + 7 + 10 + 16)         (4 + 6 + 11 + 13)

(R₁C₁, R₂C₃, R₃C_2, R₄C₄)  (R₁C₄, R₂C₂, R₃C_3, R₄C₁)

(6 + 9 + 3 + 16)         (15 + 4 + 10 + 5)

(R₂C₂, R₂C₄, R₄C_2, R₄C₄)  (R₁C₂, R₁C₄, R₃C_2, R₃C₃)

(12 + 7 + 13 + 2)         (1 + 14 + 8 + 11)

(R₂C₁, R₂C₃, R₄C_1, R₄C₃)  (R₁C₁, R₁C₃, R₃C_1, R₃C₃)
✨ Explore More Magical Patterns!
In the same way, you can discover many intriguing relationships between the numbers placed in different cells and their resulting sums. Keep exploring — every pattern reveals a new layer of mathematical beauty!

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ANIL SATPUTE