Narayan Pandit- Odd Order Magic Squares - 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 Visama magic squares (magic squares of odd order).

Fig. 7. Folding Process for Construction of 7x7 Magic Square with Sum 238

Theorem 2:³ Let n be an odd number and 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)     (12)

Now consider the square arrays S and T defined by

S(i, j) = p(i+j)   (13a)

T(i, j) = q(i j)    (13b)

Then, S and T will be magic squares whenever

p(n-1) = (n+ 1)12        (14a)

q(0) = (n-1)12  (14b)

Further, the array S+rT will be a magic square for any number r.

Proof: We first show that the square S is semi-magic. The sum of the numbers in the i-th column is given by

The sum of the numbers in the j-th row is similarly found to be the same. Now, the sum of the numbers along the diagonal i=j is given by

where the second equality follows from the fact that n is odd (2 is co-prime to n). The sum of the numbers, along the other principal diagonal i+j = n-1, is given by

n-1

p(i+j)=  p(n-1)= n(n -1)/2         (17)

i+ j =n-1           i =0

where the second equality follows from the condition (14a). Thus we have shown that S is an nxn magic square. Similarly, by making use of the condition (14b), we can show that T is also an nxn magic square. It follows that S+rT will be an nxn magic square for any number r, thereby completing the proof of Theorem 2. 

Suresh, this is a profound generalization—elevating Narāyaṇa Paṇḍita’s folding method for visama (odd-order) magic squares into a formal theorem that reveals its full generative power. Let’s unpack Theorem 2.3 and explore its mathematical structure, symbolic elegance, and creative implications for your adaptation work.

🧠 Theorem 2.3: General Construction of Odd-Order Magic Squares via Folding

🔧 Setup and Definitions

Let $nn$ be an odd integer. Define:

• $p(i)p(i)$: 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 both sequences 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(i + j)

• T(i, j) = q(i \cdot j)

These are constructed using modular arithmetic to ensure wraparound behavior—essential for toroidal symmetry and cyclic structure.

📐 Conditions for Magic Square Properties

To ensure that S and T are magic squares:

• Condition (14a):

$$p(n-1)=\frac{n+1}{2}p(n\; -\; 1)\; =\; \backslash frac\{n\; +\; 1\}\{2\}$$

• Condition (14b):

$$q(0)=\frac{n-1}{2}q(0)\; =\; \backslash frac\{n\; -\; 1\}\{2\}$$

These conditions ensure that the diagonals sum correctly and that the square is balanced across rows, columns, and diagonals.

🧮 Magic Sum and Structure

• Each row and column of S and T has the same sum

• Diagonals also sum to the same value due to the co-primality of 2 and $nn$

• The composite square $S+rTS\; +\; rT$ is a magic square for any scalar $rr$

In particular, choosing $r=nr\; =\; n$ yields a square with entries $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n^2\}\backslash \{1,\; 2,\; ...,\; n^2\backslash \}$, ensuring completeness and distinctness.

🌀 Symbolic and Structural Resonance

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

• The conditions (14a) and (14b) reflect centrality and balance—each sequence anchored at its midpoint

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

🎭 Creative Implications for Adaptation

• Permutation as Character Archetypes: Each $p(i)p(i)$ and $q(i)q(i)$ could represent a persona or motif—folded together to form a dramatic ensemble

• Folding as Ritual: The act of combining Chadya and Chadaka becomes a symbolic union—ideal for staging or visual animation

• Modular Indexing as Choreography: The wraparound behavior of the torus could be mapped to circular movement, musical phrasing, or lighting design

Would you like to co-design a generative framework that lets you explore new permutations $pp$ and $qq$ for constructing original magic squares—each with its own symbolic and dramatic flavor? We could even build a visual or interactive tool to animate the folding process and reveal its hidden symmetries.