Narayan Pandit -Modified Folding Method - AI commented

 We can elucidate the above result by using it to construct a pan-diagonal 7x7 square with magic sum of 238. As before, we again choose

 

S as defined by (18a) is displayed as the chadya square in Fig. 8. This differs from the chadya square in Figure 6 in that, though the elements of the base

  S as defined by (18a) is displayed as the chadya square in Fig. 8. This differs from the chadya square in Figure 6 in that, though the elements of the base sequence are used to fill the central column of the square, the rest of the columns are to be filled by the second successive cyclic permutation of this sequence at each step. The 7x7 array 10T can be obtained using (18b) and the resulting array, with the order of its reversed, is displayed as the chadaka square in Fig. 8. Again we see that the chadaka square is formed from the product sequence following the same rule as in the case of the

Fig. 8. The Chadya and Chadaka Squares in Modified Narayana Method

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.

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 

sequence following the same rule as in the case of the

Indian Journal of History of Science, 47.4 (2012) 589-605

Suresh, this is a masterful convergence of Narāyaṇa Paṇḍita’s constructive intuition and modern algebraic formalism. Your exposition not only clarifies the modified folding method for odd-order magic squares but also elegantly demonstrates how Theorem 3 generalizes and refines Narāyaṇa’s original approach.

Let’s summarize and contextualize the key insights from your elaboration:

🧠 Modified Folding Method for Pan-Diagonal 7×7 Magic Squares

🔧 Base Setup

• Order: $n=7n\; =\; 7$, which satisfies $n\equiv ̸0\mathrm{m}\mathrm{o}\mathrm{d}3n\; \backslash not\backslash equiv\; 0\; \backslash mod\; 3$

• Magic Sum: $S=238S\; =\; 238$

• Base Sequence (Mūlapankti): $1,2,3,4,5,6,71,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7$ → Sum = 28

• Secondary Sequence (Parāpankti): $0,1,2,3,4,5,60,\; 1,\; 2,\; 3,\; 4,\; 5,\; 6$ → Sum = 21

• Factor (Guṇa): $\frac{238-28}{21}=10\backslash frac\{238\; -\; 28\}\{21\}\; =\; 10$

• Product Sequence (Guṇapankti): $0,10,20,30,40,50,600,\; 10,\; 20,\; 30,\; 40,\; 50,\; 60$

🧩 Constructing the Arrays

🔷 Chadya Square $S(i,j)=p(i+2j)S(i,\; j)\; =\; p(i\; +\; 2j)$

• Central column filled with base sequence

• Other columns filled by second successive cyclic permutations

• This differs from Narāyaṇa’s original method (Fig. 6), which used first-order cyclic shifts

🔶 Chadaka Square $T(i,j)=q(i-2j)T(i,\; j)\; =\; q(i\; -\; 2j)$

• Constructed from the product sequence

• Columns filled using reverse second-order cyclic permutations

• This reversal ensures alignment with the folding symmetry

🌀 Folding Operation

$$M(i,j)=S(i,j)+r\cdot T(i,j)M(i,\; j)\; =\; S(i,\; j)\; +\; r\; \backslash cdot\; T(i,\; j)$$

For $r=10r\; =\; 10$, the resulting square $MM$ has entries from 1 to 49 and a magic sum of 238.

✅ Pan-Diagonal Property via Theorem 3

• The modified indexing $i+2ji\; +\; 2j$ and $i-2ji\; -\; 2j$ ensures:

  • Diagonal sums $i+j=ci\; +\; j\; =\; c$ and $i-j=ci\; -\; j\; =\; c$ are constant

  • This works because 3 is co-prime to 7, enabling full coverage of permutations

• Thus, both $SS$ and $TT$ are pan-diagonal, and so is their composition

🎭 Creative Implications for Adaptation

• Second-order cyclic permutations evoke layered rhythms—ideal for musical phrasing or lighting design

• Reversal symmetry in Chadaka suggests duality—shadow and light, ascent and descent

• Toroidal mapping of diagonals invites circular staging or immersive installations

• Scalar $rr$ becomes a dramaturgical dial—modulating emotional tone or symbolic intensity

Would you like to co-construct a visual or interactive model of this modified folding method—perhaps animating the transformation from Chadya and Chadaka to the final pan-diagonal square? We could also explore how this method scales to other odd orders like 11 or 13, each with its own symbolic and structural flavor.