Narayan Pandit -General Algebraic framework - AI commented

 We shall now see how the examples given by Narayana are particular instances of the above result. If we set n = 4 and choose p(0) = 2, p(1) = 1, p(2) = 3, p(3) = 4, q(0) = 3, q(1) = 2, q(2) = 0 and q(3) = 1, we see that the conditions (6a) and (6b) are satisfied. The resultant 4x4 array S as defined by (5a) will be

which is same as the chadya square considered by Narayana as shown in Fig. 1. The array 4x4 array 4T as defined by (5b) is given by

which is nothing but the chadaka square considered by Narayana as shown in Fig. 1, except that the order of columns is reversed. Thus the composition S+4T is nothing but the folding process of Narayana as shown in Fig. 2 and leads to the pan-diagonal 4x4 magic square as shown in the Figure.

In the same way, if we set n = 8 and choose p(0) = 4, p(1) = 3, p(2) = 2, p(3) = 1, p(4) = 5, p(5) = 6, p(6) = '7, p('7)= 8, q(0) = 7, q(1) = 6, q(2) = 5 q(3) = 4, q(4) = 0, q(5) = 1, q(6) = 2, and q(7) = 3, we see that the conditions (6a) and (6b) are satisfied. The 8x8 array S will be the same as the chadya square considered by Narayana as shown in Fig. 4. The array 8T will be the same as the chadaka square considered by Narayana as shown in Fig. 4, except that the order of columns is reversed. The composition S+8T is nothing but the folding process of Narayana as shown in Fig. 5 and leads to the pan-diagonal 8x8 magic square as shown in the Figure.

Suresh, this is a brilliant synthesis—showing how Narāyaṇa’s specific examples are not isolated marvels but instances of a general algebraic framework. You’ve elegantly bridged the intuitive folding method with its formal underpinning in modular permutations and arithmetic structure. Let’s crystallize the insight and highlight its creative and mathematical implications.

🔍 Narayāṇa’s Examples as Instances of Theorem 1.2

✅ Case 1: 4×4 Magic Square (n = 4, m = 1)

• Chosen permutations:

  • $p=[2,1,3,4]p\; =\; [2,\; 1,\; 3,\; 4]$

  • $q=[3,2,0,1]q\; =\; [3,\; 2,\; 0,\; 1]$

• Verification:

  • Condition (6a): $p(i)+p(i+2)=2+3=5p(i)\; +\; p(i\; +\; 2)\; =\; 2\; +\; 3\; =\; 5$, $1+4=51\; +\; 4\; =\; 5$ → ✅ $4m+1=54m\; +\; 1\; =\; 5$

  • Condition (6b): $q(i)+q(i+2)=3+0=3q(i)\; +\; q(i\; +\; 2)\; =\; 3\; +\; 0\; =\; 3$, $2+1=32\; +\; 1\; =\; 3$ → ✅ $4m-1=34m\; -\; 1\; =\; 3$

• Result:

  • The array $SS$ matches Narāyaṇa’s Chadya square

  • The array $4T4T$ matches the Chadaka square (column-reversed)

  • Their sum $S+4TS\; +\; 4T$ yields the pan-diagonal 4×4 magic square from Fig. 2

✅ Case 2: 8×8 Magic Square (n = 8, m = 2)

• Chosen permutations:

  • $p=[4,3,2,1,5,6,7,8]p\; =\; [4,\; 3,\; 2,\; 1,\; 5,\; 6,\; 7,\; 8]$

  • $q=[7,6,5,4,0,1,2,3]q\; =\; [7,\; 6,\; 5,\; 4,\; 0,\; 1,\; 2,\; 3]$

• Verification:

  • Condition (6a): $p(i)+p(i+4)=4+5=9p(i)\; +\; p(i\; +\; 4)\; =\; 4\; +\; 5\; =\; 9$, etc. → ✅ $4m+1=94m\; +\; 1\; =\; 9$

  • Condition (6b): $q(i)+q(i+4)=7+0=7q(i)\; +\; q(i\; +\; 4)\; =\; 7\; +\; 0\; =\; 7$, etc. → ✅ $4m-1=74m\; -\; 1\; =\; 7$

• Result:

  • The array $SS$ matches Narāyaṇa’s Chadya square in Fig. 4

  • The array $8T8T$ matches the Chadaka square (column-reversed)

  • Their sum $S+8TS\; +\; 8T$ yields the pan-diagonal 8×8 magic square in Fig. 5

🧠 Mathematical Insight

• The permutations $pp$ and $qq$ encode cyclic symmetry and complementarity

• The folding operation $S+nTS\; +\; nT$ ensures:

  • Distinct entries from 1 to $n^{2}n^2$

  • Uniform row, column, and diagonal sums

  • Pan-diagonal property via modular indexing

This shows that Narāyaṇa’s folding method is not just constructive—it’s generative, capable of producing an entire class of pan-diagonal magic squares for any $n=4mn\; =\; 4m$.

🎭 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.