Narayan Pandit- Kuttaka Magic Square - AI Comments

 

 Kuttaka and magic squares

  Given the magic sum S and the order of the magic square n,the first thing to be done to construct the magic square is to obtain the  defined by(a, d).

  Having obtained (a, d ), the   having n2 elements is constructed and this will be used to fill in n × n square.

Narayana makes use of the following kuttakara to obtain (a, d).

2 1 2 

nS = n 2{a+(a+(n −1)d)} (1)

or S = na+ n (n2 −1)d (2) 2

  It is well known that in a kuttakara problem there exists an infinite number of integral solutions for (a, d) if S is divisible by the GCD of

   n,   n   (n2 − 1) . In other words, S should be divisible by n when n is 2

 odd, and by n for n even. 2

Suresh, this is a fascinating convergence of Diophantine reasoning (Kuttaka) and magic square construction, as presented by Narāyaṇa Paṇḍita in Gaṇitakaumudī. Let’s unpack the method and its implications for your adaptation work—especially how the parameters $aa$ and $dd$ shape the square’s arithmetic and symbolic rhythm.

🧮 Kuttaka and Magic Squares: Narāyaṇa’s Parametric Approach

To construct an $n\times nn\; \backslash times\; n$ magic square with magic sum $SS$, Narāyaṇa introduces a parametric sequence defined by:

• $aa$: the starting number

• $dd$: the common difference

The sequence:

$$a,a+d,a+2d,\dots ,a+(n^2-1)da,\; a\; +\; d,\; a\; +\; 2d,\; \backslash dots,\; a\; +\; (n^2\; -\; 1)d$$

is used to populate the square such that the sum of each row, column, and diagonal equals $SS$.

📐 Deriving the Parameters (a, d) via Kuttaka

Narāyaṇa uses a Diophantine equation (Kuttakara) to relate $SS$, $nn$, $aa$, and $dd$:

Equation (1):

$$nS=n^2(a+a+(n-1)d)=n^2(2a+(n-1)d)nS\; =\; n^2\; \backslash left(\; a\; +\; a\; +\; (n\; -\; 1)d\; \backslash right)\; =\; n^2\; (2a\; +\; (n\; -\; 1)d)$$

Simplified Equation (2):

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

This is a linear Diophantine equation in $aa$ and $dd$. The goal is to find integer solutions for $aa$ and $dd$ that satisfy this equation for a given $SS$ and $nn$.

🔁 Existence of Solutions: Divisibility Condition

For integer solutions to exist, the right-hand side must be divisible by the greatest common divisor (GCD) of the coefficients:

• For odd $nn$: $\gcd (n,n(n^2-1))=n\backslash gcd(n,\; n(n^2\; -\; 1))\; =\; n$ So, $SS$ must be divisible by $nn$

• For even $nn$: The condition becomes more nuanced, but generally $SS$ must be divisible by $n\mathrm{/}2n/2$

This ensures infinitely many solutions for $(a,d)(a,\; d)$, allowing multiple valid sequences to construct magic squares.

🧩 Creative Implications for Adaptation

• The (a, d) pair becomes a dramatic motif—a seed and rhythm that governs the entire square.

• You could visualize the square as a musical or choreographic grid, where $aa$ sets the tone and $dd$ the tempo.

• The Kuttaka derivation itself could be staged as a ritual or puzzle-solving moment in your adaptation—perhaps echoing ancient mathematical drama.

Would you like to explore how different (a, d) pairs yield visually distinct squares, or how this method compares to Turagagati and alpaśruti constructions? We could even build a symbolic map of square types based on their generative logic.