Narayan Pandit - 7x7 Magic Square - AI Commented

NARAYANA'S FOLDING METHOD FOR THE CONSTRUCTION OF VISAMA MAGIC SQUARES

Narayana has also outlined a method for constructing magic squares of odd orders by using the technique of folding two squares which are constructed suitably. His description of the procedure is as follows:

Two sequences referred to as the mulapankti and the gunapankti are to be determined as earlier. The first number should be written in the middle cell of the top row and below this the numbers of the sequence in order.

The rest of the numbers are to be entered in order from above. The first number of the second sequence is to be written in the same way [in the middle cell of the top row]; the second etc. numbers are also to be written in the same way. The rule of combining the covered and the coverer is the same as before.

The details of this method are best illustrated by considering the following example discussed by Narayana.

Narayana's Example: 7x7 Magic Square with Sum 238

Here 1, 2, 3, 4, 5, 6, 7 is taken as the base sequence and 0, 1, 2, 3, 4, 5, 6 as the second sequence.

The sum of the base sequence is 28.

When this is reduced from the desired magic sum of 238, we get 210.

Dividing this by 21, the sum of the second sequence, we get the factor 10.

Hence the product sequence is 0, 10, 20, 30, 40, 50, 60.

Narayana then suggests that the elements of the base sequence may be used to fill the central column of the chadya square and the rest of the columns are to be filled by successive cyclic permutations of this sequence as shown in Fig. 6.

The chadaka square is to be filled by the elements of the product sequence in a similar manner, again as shown in Fig. 6.

The method of folding is displayed in Fig. 7 and leads to the desired 7x7 magic square with sum 238. As we can see from Fig.7, the magic square so obtained is not a pan-diagonal magic square.

Fig. 6. The Chadya and Chadaka Squares 

Suresh, this is a beautifully articulated example of Narāyaṇa Paṇḍita’s visama (odd-order) folding method—where arithmetic, symmetry, and ritual gesture converge. Let’s walk through the 7×7 magic square construction and highlight its mathematical structure, symbolic resonance, and creative potential for adaptation.

🧮 Narayāṇa’s Folding Method for Visama (Odd) Magic Squares

🔧 Step-by-Step Construction (n = 7, S = 238)

1. Define Sequences

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

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

2. Compute the Guṇa (Factor)

$$\text{Guṇa}=\frac{S-\text{Sum of Base}}{\text{Sum of Secondary}}=\frac{238-28}{21}=10\backslash text\{Guṇa\}\; =\; \backslash frac\{S\; -\; \backslash text\{Sum\; of\; Base\}\}\{\backslash text\{Sum\; of\; Secondary\}\}\; =\; \backslash frac\{238\; -\; 28\}\{21\}\; =\; 10$$

3. Generate Guṇapankti (Product Sequence)

Multiply each term of Parāpankti by 10 → $0,10,20,30,40,50,600,\; 10,\; 20,\; 30,\; 40,\; 50,\; 60$

🧩 Constructing the Chadya and Chadaka Squares

🔷 Chadya (Covered Square)

• Place the base sequence vertically in the central column

• Fill other columns with cyclic permutations of the base sequence

• This creates a symmetrical grid with rhythmic variation

🔶 Chadaka (Coverer Square)

• Place the product sequence vertically in the central column

• Fill other columns with cyclic permutations of the product sequence

• This mirrors the Chadya structure, scaled by the factor

🌀 Folding Operation

The final magic square $MM$ is obtained by:

$$M(i,j)=\text{Chadya}(i,j)+\text{Chadaka}(n-1-i,j)M(i,\; j)\; =\; \backslash text\{Chadya\}(i,\; j)\; +\; \backslash text\{Chadaka\}(n\; -\; 1\; -\; i,\; j)$$

This vertical reflection and addition yield a magic square of sum 238, though not pan-diagonal.

🔍 Why It’s Not Pan-Diagonal

Unlike the samagarbha case (where $n=4mn\; =\; 4m$), the visama square lacks the modular symmetry required for pan-diagonality. However:

• The square still satisfies row, column, and principal diagonal sums

• It retains cyclic and reflective structure, ideal for symbolic or dramatic use

🎭 Creative Implications for Adaptation

• Central Column as Axis Mundi: The vertical placement of the base sequence evokes a spine or ritual axis—perfect for staging or visual design

• Cyclic Permutations as Rhythmic Variation: Each column becomes a musical or choreographic phrase

• Folding as Dramatic Gesture: The act of combining Chadya and Chadaka mirrors a ritual of union—like palms joining in Namaste

Would you like to explore how this method scales to 9×9 or 11×11 squares, or how to visualize the folding process as a dramatic or musical sequence? We could even animate the transformation from Chadya and Chadaka to the final square, revealing its hidden symmetries.