Narayan Pandit-Folding Method for Magic Square - AI Comments

  In 1938, Rosser and Walker proved that this is in fact the exact number of 4x4 pan-diagonal magic squares with entries 1, 2, ..., 16. 

Vijayaraghavan (1941) gave a much simpler proof of this result, which have been outlined by Sridharan and Srinivas (2011, pp.389-391), 

Narayana goes on to discuss general methods of construction of magic squares depending upon whether the square is samagarbha (doubly-even), visamagarbha (singly-even) or visama (odd). In the case of samagarbha and visama squares, apart from discussing the traditionally well-known methods of construction (indicated for instance in Pheru's work earlier), 

Narayana presents an entirely new method known as samputa-vidhi (method of folding). This is a general method of construction of magic squares by composing or folding two magic squares constructed suitably

Narayana's Folding Method for the Construction of Samagarbha Magic squares

Narayana's samputavidhi (folding method) involves the construction of two auxiliary magic squares, which are called the chadya (covered) and chadaka (coverer). As he states (see below), the process of folding involves covering of the chadya by the chadaka like in the folding of the palms.

In what follows we shall adopt the convention that the columns (rows) of an nxn square array are indexed from the left (top) by the integers 0,1,2...,n-1. We shall denote the element at the intersection of the i-th column and j-th row of the array M by M(i,j). Now, if M₁ and M2 are two nxn square arrays, then the process of folding results in the square array M, whose (i,j)- th element is given by

M(ij) = M1(ij) + M₂(n-1-i,j)        (³)

for all

Suresh, this is a beautifully layered synthesis of mathematical history, algorithmic elegance, and symbolic geometry. Let’s distill the essence of Narāyaṇa Paṇḍita’s Samputavidhi (folding method) and its place in the lineage of magic square construction—while also connecting it to the modern proofs and classifications you’ve cited.

🧠 Historical Validation: The 384 Pan-Diagonal 4×4 Magic Squares

• Rosser & Walker (1938): Provided a rigorous combinatorial proof that exactly 384 distinct 4×4 pan-diagonal magic squares exist using the numbers 1–16.

• T. Vijayaraghavan (1941): Offered a simpler and more elegant proof, later outlined by Sridharan & Srinivas (2011) in their historical survey.

• These results validate Narāyaṇa’s empirical insight from Gaṇitakaumudī, where he displayed 24 such squares and noted the possibility of generating 384 via cyclic permutations.

🧩 Samputavidhi: Narāyaṇa’s Folding Method

This method is a constructive algorithm for generating magic squares, especially of samagarbha (doubly-even) and visama (odd) types.

🔧 Construction Steps:

1. Create Two Auxiliary Squares:

   • Chadya (Covered): Often derived from a base arithmetic sequence

   • Chadaka (Coverer): Derived from a transformed or scaled sequence

2. Indexing Convention:

   • Columns and rows indexed from $00$ to $n-1n-1$

   • Element at position $(i,j)(i,\; j)$ in square $MM$ is denoted $M(i,j)M(i,\; j)$

3. Folding Rule:

$$M(i,j)=M_1(i,j)+M_2(n-1-i,j)M(i,\; j)\; =\; M\_1(i,\; j)\; +\; M\_2(n\; -\; 1\; -\; i,\; j)$$

• This reflects a vertical folding: the Chadaka is flipped top-to-bottom and added to the Chadya

• The result is a new square $MM$ that satisfies the magic sum condition

🌀 Symbolic Interpretation:

• The folding mimics palms joined in prayer—a gesture of symmetry, union, and balance

• The Chadya and Chadaka represent complementary forces—like prakṛti and puruṣa, or shadow and light

🧮 Classification of Squares in Narāyaṇa’s Framework

Type	Definition	Folding Outcome
Samagarbha	Doubly-even (e.g., 4×4, 8×8)	Always yields pan-diagonal squares
Visamagarbha	Singly-even (e.g., 6×6)	Requires modification for symmetry
Visama	Odd (e.g., 3×3, 5×5)	Not pan-diagonal by default; modifiable if not divisible by 3

This typology allows Narāyaṇa to tailor his folding method to the structural nature of the square, offering both generality and elegance.

🎭 Creative Adaptation Potential

• The folding equation $M(i,j)=M_1(i,j)+M_2(n-1-i,j)M(i,j)\; =\; M\_1(i,j)\; +\; M\_2(n-1-i,j)$ could be dramatized as a ritual of convergence—two grids merging to form cosmic order.

• You might visualize Chadya and Chadaka as light and shadow grids, with their union producing a mandala of balance.

• The indexing and folding could be staged as choreographic movements, echoing mathematical precision and spiritual symbolism.

Would you like to co-construct a visual or performative treatment of this folding method—perhaps animating the grid transformation or mapping it to musical phrasing? We could also explore how this method scales to 8×8 or 6×6 squares, adapting Narāyaṇa’s logic to modern digital formats.