Narayan Pandit - 7x7 Magic Square by Folding Method - AI commented

 We shall now see how the example given by Narayana is a particular instance of the above result. If we set n = 7 and choose p(0) = 5, p(1) = 6,

which is same as the chadya square considered by Narayana as shown in Fig. 6. The array 7x7 array 10T as defined by (13b) is given by

which is nothing but the chadaka square considered by Narayana as shown in Figure 6, except that the order of columns is reversed. Thus the composition S+10T is nothing but the folding process of Narayana as shown in Fig.7 and leads to the 7x7 magic square as shown in the Figure.

As we noted earlier, the 7x7 magic square in Fig. 7 obtained by Narayana's folding method is not pan-diagonal. We shall now show that, for the case of odd-numbers n which are not multiples of 3 (that is, numbers of the form 6m±1), a simple modification of Narayana's folding method can be used to construct pan-diagonal magic squares.

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.