Narayan Pandit - Theorem 3 Magic Squares - AI Commented

 Theorem 3: Let n be an odd number not divisible by 3, and let p(i) for 1 i n and q(i) for 0 i n-1 be as defined in Theorem 2, satisfying (12), (14a) and (14b). Then, the square arrays S and T defined by

S(i, j) = p(i+2j) (18a)

T(i, j) = q(i-2j)  (18b)

will be nxn pan-diagonal magic squares and the same is true of the array S+rT for any number r.

Proof: We first consider the array S defined by (18a). It is easy to see that S is a semi-magic square, following the same line of argument used in Theorem 2. To show that it is pan-diagonal, we consider the sums along the diagonals i+j = c, for 0 c n-1, and obtain

n-1p(i+2j)=      p(j +c) = n (n-1)12       (19)

i+ j =c. i =0

We now consider the sum along the diagonals i-j = c, for 0 c n-1, and obtain 

where the second equality follows from the fact that 3 is co-prime to n. The fact that T is an nxn pan-diagonal magic square can be proved along the same lines. It then follows that that S+rT will be an nxn pan-diagonal magic square for any number r, thereby completing the proof of Theorem 3.

Suresh, this is a remarkable culmination—Theorem 3 elevates Narāyaṇa Paṇḍita’s folding method into a universal generator for pan-diagonal magic squares of odd order, provided the order $nn$ is not divisible by 3. Let’s unpack the structure, logic, and creative implications of this theorem with clarity and reverence.

🧠 Theorem 3: Pan-Diagonal Magic Squares for Odd $n\equiv ̸0\mathrm{m}\mathrm{o}\mathrm{d}3n\; \backslash not\backslash equiv\; 0\; \backslash mod\; 3$

🔧 Setup and Definitions

Let:

• $nn$ be an odd integer not divisible by 3

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

• $q(i)q(i)$: 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 + 2j)

• T(i, j) = q(i − 2j)

These are constructed using modular arithmetic to ensure wraparound behavior—essential for toroidal symmetry and pan-diagonal properties.

📐 Conditions for Magic Square Properties

To ensure pan-diagonality:

• 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 anchor the sequences at their midpoints, ensuring diagonal balance.

🧮 Proof Sketch and Key Insights

✅ Diagonal $i+j=ci\; +\; j\; =\; c$:

$$\sum _{i+j=c}p(i+2j)=\sum _{j=0}^{n-1}p(j+c)=\text{constant}\backslash sum\_\{i+j=c\}\; p(i\; +\; 2j)\; =\; \backslash sum\_\{j=0\}^\{n-1\}\; p(j\; +\; c)\; =\; \backslash text\{constant\}$$

This works because 2 is co-prime to $nn$, ensuring full coverage of the permutation.

✅ Diagonal $i-j=ci\; -\; j\; =\; c$:

$$\sum _{i-j=c}p(i+2j)=\sum _{j=0}^{n-1}p((1+2)j+c)=\text{constant}\backslash sum\_\{i-j=c\}\; p(i\; +\; 2j)\; =\; \backslash sum\_\{j=0\}^\{n-1\}\; p((1\; +\; 2)j\; +\; c)\; =\; \backslash text\{constant\}$$

This works because 3 is co-prime to $nn$—a critical condition that fails when $n\equiv 0\mathrm{m}\mathrm{o}\mathrm{d}3n\; \backslash equiv\; 0\; \backslash mod\; 3$.

Thus, both S and T are pan-diagonal magic squares, and so is their composition:

$$S+rTS\; +\; rT$$

for any scalar $rr$, including $r=nr\; =\; n$, which yields a square with entries $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n^2\}\backslash \{1,\; 2,\; ...,\; n^2\backslash \}$.

🌀 Symbolic and Structural Resonance

• The use of 2j and −2j in indexing reflects duality and inversion—a dance of symmetry across the grid

• The co-primality conditions evoke cosmic harmony—only when the structure avoids divisibility by 3 does full symmetry emerge

• The folding method becomes a modular ritual, where permutations are layered to produce balance and completeness

🎭 Creative Implications for Adaptation

• Diagonal indexing as movement: The $i+2ji\; +\; 2j$ and $i-2ji\; -\; 2j$ patterns could be mapped to choreographic or musical phrasing

• Scalar $rr$ as emotional modulation: Each value of $rr$ yields a different dramatic tone—like shifting key or lighting

• Toroidal symmetry: The pan-diagonal property invites circular staging, wraparound visuals, or immersive installations

Would you like to co-design a generative framework to explore these constructions interactively—perhaps animating the folding process or mapping it to dramatic motifs? We could even build a symbolic grid that visualizes how co-primality governs symmetry and balance.