Narayan Pandit -Magic Squares of ordr n=4m AI Commented

 We shall now show that, Narayana's folding method is indeed a general procedure, which can be used to construct a large class of pan-diagonal samagarbha magic squares (magic squares of order 4m).

Theorem 1:² Let n=4m be a number divisible by 4. Let p(i), for 1 i n, denote an arbitrary permutation of 1, 2, ..., n and let q(i), for 0 i n-1, denote an arbitrary permutation of 0, 1, ..., n-1. We can define p(i) and q(i) for all integers i by assuming that

p(i) = p(k) and q(i) = q(k) whenever is "k (mod n)      (4)

Now consider the square arrays S and T defined by

S(i, j) = p(j +2mi)         (5a)

T(i, j) = q(i+2mj)          (5b)

Then, S and T will be pan-diagonal magic squares whenever

p(i) + p(i+2m) = 4m+1 (6a)

q(i) + q(i+2m) = 4m-1  (6b)

Further, the array S+rT will be a pan-diagonal magic square for any number r; in particular S+nT will be a pan-diagonal magic square with entries 1, 2, ...n².

Proof: We first show that the sums of all the rows and principal diagonals of T are the same. The sum of the numbers in the j-th row is

4m-1    4m-1

q(i+2mj) =       q(i) = (4m) (4m-1)/2    (⁷)

i =0      i=0

The sum of the numbers in the diagonal i=j are given by

4m-1    4m-1    4 Ill- 1

q(i+2mj) =       q[(2m+1)i] =    q(i) =(4m) (4m-1)/2 ...(8)

i= j =0  i =0      i =0

where we have used the fact that (2m+1) is co-prime to 4m. The sum of the numbers in the other principal diagonal i+j = 4m-1 is given by

4m-1    4m-1

q(i+2mj) =       q(i +2m(4m-1-i)) =      q[i(1-2m) - 2m] =(4m) (4m-1)/2

i+ j -4m-1         i =0      i =0

(⁹)

where we have used the fact that (1-2m) is co-prime to 4m.

We shall now see that the condition 6(b) is needed in order to ensure that each column of T also has the same magic sum (4m)(4m-1)/2. The sum of the numbers in the i-th column is given by

4m-1    2m-1

I q(i+2inj)=       [q(i+4mj) + q(i+4mj+2m)] = (4m) (4m-1)/2

j =0      j =0 

where we have made use of the condition (6b).

Thus we have shown that T is a magic square. To show that it is pan-diagonal, we consider the sums along the diagonals i+j = c, for 0 c 4m1, and obtain

4m-1    4m-1

q(i+2mj)=         q[i +2m(c-i)]     q[i(1-2m) +Inc] =(4m) (4m-1)/2

i+ j =c  i =0      i =0

 where we have used the fact that (1-2m) is co-prime to 4m. That the other set of diagonals i-j = c, for 0 c 4m-1, also add to the same sum, can be proved along the same lines. 

Thus we have shown that T is a pan-diagonal magic square. In the same way, it can be shown that S is also a pan-diagonal magic square, whenever (6a) is satisfied. Hence, it follows that S+rT will be a pan-diagonal magic square for any number r. The fact that S+nT has entries 1, 2, ...n² can be shown by a simple argument which demonstrates that no two elements of the array S+nT are the same. This completes the proof Theorem 1.

Suresh, this is a magnificent formalization—elevating Narāyaṇa Paṇḍita’s samputavidhi from an elegant construction to a general theorem for generating pan-diagonal magic squares of order $n=4mn\; =\; 4m$. Let’s distill the structure, implications, and creative potential of Theorem 1.2, while preserving its mathematical rigor and symbolic richness.

🧠 Theorem 1.2: General Construction of Pan-Diagonal Magic Squares of Order $n=4mn\; =\; 4m$

🔧 Setup and Definitions

Let:

• $n=4mn\; =\; 4m$, a doubly-even number

• $p(i)p(i)$: a permutation of $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}\backslash \{1,\; 2,\; ...,\; n\backslash \}$

• $q(i)q(i)$: a permutation of $\{0,1,\mathrm{.}\mathrm{.}\mathrm{.},n-1\}\backslash \{0,\; 1,\; ...,\; n-1\backslash \}$

Extend $p(i)p(i)$ and $q(i)q(i)$ periodically:

$$p(i)=p(i\mathrm{m}\mathrm{o}\mathrm{d}n),q(i)=q(i\mathrm{m}\mathrm{o}\mathrm{d}n)p(i)\; =\; p(i\; \backslash mod\; n),\; \backslash quad\; q(i)\; =\; q(i\; \backslash mod\; n)$$

Define two square arrays:

• S(i, j) = p(j + 2mi)

• T(i, j) = q(i + 2mj)

These are structured using modular arithmetic to ensure wraparound behavior—essential for toroidal symmetry.

📐 Conditions for Pan-Diagonality

To ensure that S and T are pan-diagonal magic squares:

• Condition (6a):

$$p(i)+p(i+2m)=4m+1p(i)\; +\; p(i\; +\; 2m)\; =\; 4m\; +\; 1$$

• Condition (6b):

$$q(i)+q(i+2m)=4m-1q(i)\; +\; q(i\; +\; 2m)\; =\; 4m\; -\; 1$$

These conditions enforce complementarity across diagonals—each pair sums to a constant, ensuring uniformity across rows, columns, and diagonals.

🧮 Magic Sum and Structure

• Each row, column, and diagonal of T sums to:

$$\frac{n(n-1)}{2}=\frac{4m(4m-1)}{2}\backslash frac\{n(n\; -\; 1)\}\{2\}\; =\; \backslash frac\{4m(4m\; -\; 1)\}\{2\}$$

• The same applies to S, with its own permutation structure.

Thus, S + rT is a pan-diagonal magic square for any scalar $rr$. In particular:

• S + nT yields a square with entries $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n^2\}\backslash \{1,\; 2,\; ...,\; n^2\backslash \}$

• This ensures distinctness and completeness of the square

🌀 Symbolic and Structural Resonance

• The folding method becomes a modular composition—two grids interwoven via arithmetic and symmetry

• The conditions (6a) and (6b) reflect duality: each element paired with its complement

• The periodicity and modular indexing evoke cyclical time, ritual recurrence, and cosmic order

🎭 Creative Implications for Adaptation

• Visual Treatment: Animate the folding of S and T—perhaps as rotating mandalas or mirrored grids

• Musical Mapping: Let $p(i)p(i)$ and $q(i)q(i)$ define melodic and rhythmic sequences, merged via scalar modulation

• Dramatic Structure: Use the scalar $rr$ as a metaphor for transformation—each value yielding a new emotional or narrative tone

• Educational Visualization: Build interactive grids showing how permutations and modular shifts yield pan-diagonal harmony

Would you like to co-design a symbolic visualization of this theorem—perhaps mapping the folding process to stage movement, lighting, or musical phrasing? We could also explore how this general method scales to 8×8, 12×12, or even 16×16 squares, each with its own dramatic and pedagogical potential.