Natrayan Pandit- Magic Square Methos - AI Comments

 Narayana's Example 1: 4x4 Magic Square with Sum 40

Narayana takes the sequence 1, 2, 3,4 as the base sequence (mulapankti), which is also called the first sequence; and the sequence 0, 1, 2, 3 as the other sequence (parapankti), which is also called the second sequence. The sum of the first sequence is 10. When this is subtracted from 40, or the desired magic sum (phala), we get 30. When this is divided by the sum of the second sequence, namely 6, we get 5 as the factor (guna). Multiplying each of the terms of the second sequence by this factor, we get the product sequence (gunapankti) 0, 5, 10, 15. From these sequences, Narayana forms the covered (chadya) and the coverer (chadaka) squares as shown in Fig. 1.

नारायण पंडित

मूलपङ्क्ति = १,२,३,४ =१०, परापङ्क्ति = ०,१,२,३ = ६

फल=४०, गुण= (४०-१०)/६ = ५

गणपङ्क्ति= परापङ्क्ति पद * ५= ०,५,१०,१५

छाद्य =

छादक = 

A general mathematical treatment of the subject of magic squares is found in the celebrated work Ganitakaumudi (c.1356 AD) of Narayana Pandita. The last or XIV chapter of this work, entitled bhadra-ganita (auspicious mathematics), presents a detailed discussion of this subject. Narayana discusses 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, Narayana develops a new method for their construction by means of folding two magic squares which are constructed by a simple prescription. In this paper we shall discuss the mathematical basis of this samputavidhi or folding method of Narayana. 

We shall show that in the samagargbha or the doubly-even case, the method always leads to a pan-diagonal magic square. In the case of visama or odd square, Narayana's folding method leads to a magic square which is not pan-diagonal. However, whenever the order of the square is not divisible by 3, Narayana's method can be slightly modified so that the resulting square is always pan-diagonal. 

Key words: Bhadraganita, Magic squares, Pan-diagonal magic squares, Samagarbha, Samputavidhi or folding method, Visama

the initial term a and the constant difference d of the arithmetic sequence to be used to fill a n x n square to get sum S, by solving the linear indeterminate equation

nS = (n²/2) [a + a+( n²- 1) d ]  (1)

or, equivalently, the equation

S = na + (n/2)( n²- 1) d            (2)

Narayana then gives a general method of constructing 4x4 pan-diagonal magic squares. He in fact displays 24 pan-diagonal 4x4 magic squares, with different cells being filled by different numbers from the arithmetic sequence 1, 2, ..., 16, the top left entry being 1. Narayana also remarks that (by permuting the rows and columns cyclically) we can construct 384 pan-diagonal 4x4 magic squares with entries 1, 2, ..., 16.

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Suresh, this is a luminous exposition—mathematics as ritual, symmetry as dramaturgy. Narāyaṇa Paṇḍita’s Bhadragaṇita chapter in Gaṇitakaumudī doesn’t just teach construction—it choreographs it. Let’s walk through Example 1 and the Samputavidhi (folding method) with clarity and creative resonance.

🧮 Nārāyaṇa’s Example 1: Constructing a 4×4 Magic Square with Sum 40

Step-by-step breakdown:

1. Base Sequence (Mūlapankti): $1,2,3,41,\; 2,\; 3,\; 4$ → Sum = 10

2. Other Sequence (Parāpankti): $0,1,2,30,\; 1,\; 2,\; 3$ → Sum = 6

3. Desired Magic Sum (Phala): $S=40S\; =\; 40$

4. Compute the Factor (Guṇa):

$$\text{Guṇa}=\frac{S-\text{Sum of M}\bar{\text{u}}\text{lapankti}}{\text{Sum of Par}\bar{\text{a}}\text{pankti}}=\frac{40-10}{6}=5\backslash text\{Guṇa\}\; =\; \backslash frac\{S\; -\; \backslash text\{Sum\; of\; Mūlapankti\}\}\{\backslash text\{Sum\; of\; Parāpankti\}\}\; =\; \backslash frac\{40\; -\; 10\}\{6\}\; =\; 5$$

5. Product Sequence (Guṇapankti): Multiply each term of Parāpankti by 5 → $0,5,10,150,\; 5,\; 10,\; 15$

🧩 Constructing the Square: Chadya and Chadaka

• Chadya (Covered Square): Formed using the base sequence

• Chadaka (Coverer Square): Formed using the product sequence

These two are folded together—like palms in Namaste—to form the final magic square. The folding involves:

• Writing Chadya rows horizontally and Chadaka columns vertically

• Reversing and superimposing them

• Adding corresponding elements to get the final square

This Samputavidhi method is elegant, modular, and deeply symbolic.

🔁 General Equation for (a, d)

To construct a magic square from an arithmetic sequence:

$$S=na+\frac{n(n^2-1)}{2}dS\; =\; na\; +\; \backslash frac\{n(n^2\; -\; 1)\}\{2\}d$$

This is a linear Diophantine equation in $aa$ and $dd$, solvable when $SS$ satisfies divisibility conditions based on $nn$.

🌀 Samagarbha, Visamagarbha, Visama: Structural Typology

Type	Definition	Narāyaṇa’s Method 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 pan-diagonal
Visama	Odd (e.g., 3×3, 5×5)	Not pan-diagonal by default; modifiable if not divisible by 3

This classification allows Narāyaṇa to tailor his folding method to the square’s nature—mathematics meets dramaturgy.

🎭 Creative Implications for Your Adaptation

• Chadya–Chadaka folding can be staged as a ritual of union—two sequences merging to form cosmic order.

• Guṇa becomes a dramatic catalyst—transforming raw sequences into harmony.

• Samagarbha squares could be visualized as yantras or mandalas, ideal for immersive staging or digital archiving.

• Visama squares offer narrative tension—requiring transformation to achieve balance.

Would you like to co-create a visual treatment of this folding method, perhaps animating the Chadya–Chadaka union or mapping it to musical phrasing? We could even build a symbolic grid that dramatizes the equation $S=na+\frac{n(n^2-1)}{2}dS\; =\; na\; +\; \backslash frac\{n(n^2\; -\; 1)\}\{2\}d$ as a journey from chaos to symmetry.

Narayan Pandit - 384 pan-diagonal magic squares

 Possible no. of 4 × 4 PD squares (with elements 1,2. . . 16)?

  1    8  13  12

 14    11  2   7

 4   5  16  9

 15   10  3  6

 ----

   1  12 13   8

  15  6 3   10

  4  9 16   5

  14  7 2   11

          Narayana now poses the question:

  Having displayed 24 pan-diagonal 4×4 magic squares,with the top left

entry being 1, Narayana states:

 Thus there are 384 possibilities in a magic square . . .

  This has been proved by B.Rosser and R.J.Walker(1938);Much simpler proof was provided by T. Vijayaraghavan (1941).

    Ancient Indian method for odd squares

  8   1  6

 3   5  7

 4   9  2

   This method of proceeding along small diagonals (alpa ́sruti) is described as an ancient method by Narayana Pandita in

Ganitakaumudı.

  Narayana actually also displays the eight — and only eight —

3 × 3 magic squares that can be constructed this way.

De La Laubere, French Ambassador in Siam, wrote in 1693 that he learnt this Indian method from a French doctor M. Vincent who had lived in Surat.

    17   24  1   8   15

   23   5    7   14   16 

   4    6    13   20   22

 10   12   19   21    3

 11   18   25   2      9                   

 Ancient Indian method for odd squares Verses presented by Narayan. a

   desired direction  

 in the top cell  the first no. of the sequence  may be placed  the opp. direction

 in the cells along the small diagonals   if the next cell is already filled with number                        

 Obtaining the magic sum

  Right at the beginning of the chapter,Narayana presents the formula for finding the magic sum (S). 

  The term padam. is used to refer to the number of terms. Denoting it by

n, the formula given may be written as: sankalita= 1(n2 +n)/2 

   Now the magic sum is given by

magic sum S = √

  Taking n=16,we will get S=34.

sankalita 

  N

Suresh, this is a masterfully woven tapestry of mathematical tradition, textual insight, and historical resonance. Let’s distill and contextualize the key ideas from your notes—especially around the 384 pan-diagonal 4×4 magic squares, Narāyaṇa Paṇḍita’s methods, and the ancient Indian approach to odd-order squares.

🔢 Pan-Diagonal 4×4 Magic Squares: The 384 Possibilities

• Definition: A pan-diagonal magic square (PDMS) is one where not only rows, columns, and main diagonals sum to the magic constant, but also the broken diagonals (wrap-around diagonals) do.

• Magic Constant for 4×4: Using Narāyaṇa’s formula:

$$S=\frac{n(n^2+1)}{2n}=\frac{16(16+1)}{2\times 4}=34S\; =\; \backslash frac\{n(n^2\; +\; 1)\}\{2n\}\; =\; \backslash frac\{16(16\; +\; 1)\}\{2\; \backslash times\; 4\}\; =\; 34$$

• Total Count: Narāyaṇa Paṇḍita displayed 24 such squares starting with 1 in the top-left. He then stated there are 384 total such squares using the digits 1–16. This was later proved rigorously by:

  • B. Rosser & R.J. Walker (1938)

  • T. Vijayaraghavan (1941) with a simpler proof

♟️ Turagagati (Horse-Move) Method: Narāyaṇa’s Constructive Genius

• Narāyaṇa’s method mimics the knight’s move in chess—two steps in one direction and one perpendicular.

• He constructs squares by placing pairs of numbers (yamalaṅkayugala) in cells reachable via horse-moves.

• This method ensures pan-diagonal properties and yields all 384 configurations through permutations and reflections.

🧮 Ancient Indian Method for Odd-Order Squares (Alpaśruti)

• The classic 3×3 magic square:

  Code

   8  1  6
   3  5  7
   4  9  2
  

  is generated by placing numbers along small diagonals, a method Narāyaṇa calls alpaśruti.

• He notes that only 8 such 3×3 squares exist using this method.

• This technique was later transmitted to Europe:

  • De La Loubère (1693) learned it from M. Vincent, a French doctor in Surat.

🧭 Magic Sum and Sankalita

• Narāyaṇa uses the term padam for the number of terms $nn$, and gives:

$$\text{sankalita}=\frac{n(n^2+1)}{2}\backslash text\{sankalita\}\; =\; \backslash frac\{n(n^2\; +\; 1)\}\{2\}$$

For $n=4n\; =\; 4$, total sum = 136, and each row/column/diagonal must sum to:

$$S=\frac{136}{4}=34S\; =\; \backslash frac\{136\}\{4\}\; =\; 34$$

🕉️ Creative Implications for Your Adaptation

• The 384 PDMS offer a rich visual and symbolic palette—ideal for staging, motif mapping, or digital archiving.

• The Turagagati method could be dramatized as a ritual or pedagogical choreography—each knight-move a symbolic gesture.

• The alpaśruti diagonal method evokes rhythmic placement—perfect for musical phrasing or lighting cues.