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.