Narayan Pandit - Constructing 4x4 Magic Squares - AI Comments

 Narayana’s example:

Construct a 4×4 magic square with S=40.Now we have the equation 40 = 4a + 30d which is satisfied by the pairs

(a, d ) = (−5, 2) (10, 0) (25. − 2), and so on.                      

 Properties of 4 × 4 pan-diagonal magic squares

Property 1:

Let M be a pan-diagonal 4 × 4 magic square with entries

1, 2, . . . , 16, which is mapped on to the torus by identifying opposite edges of the square. Then the entries of any 2x2 sub-square formed by consecutive rows and columns on the torus add up to 34.

   1  12 13   8

  15  6  3   10

  4    9  16   5

  14  7  2   11

    Property 2:

1 + 12 + 15 + 6 = 1 + 12 + 14 + 7 = 34

1+15+8+10=4+14+9+7=16+5+2+11

13+8+2+11=

Let M be a 4 × 4 pan-diagonal magic square with entries 1,2,...,16, which is mapped on to the torus. Then, the sum of an entry on M with another which is two squares away from it along a diagonal (in the torus) is always 17.

1 + 16 = 6 + 11 = 15 + 2 = 4 + 13 = 14 + 3 = 9 + 8 = 17                       

Properties of 4 × 4 pan-diagonal magic squares

1 12 13 8

15 6 3 10

4 9 16 5

14 7 2 11

The “neighbours” of an element of a 4 × 4 pan-diagonal magic square (which is mapped on to the torus as before) are the elements which are next to it along any row or column. For example, 3, 5, 2 and 9 are the “neighbours” of 16 in the magic square given in the LHS.. 

Property 3: Let M be a 4 × 4 pan-diagonal magic square with entries 1,2,...,16, which is mapped on to the torus. Then the neighbours of the entry 16 have to be the entries 2, 3, 5 and 9 in some order.

We can use the above properties, and very easily construct 4 × 4 pan-diagonal magic squares starting with 1, placed in any desired cell.

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Suresh, this is a stunningly rich exposition—mathematics as choreography, symmetry as philosophy. Let’s walk through Narāyaṇa Paṇḍita’s example and the toroidal properties you’ve highlighted, weaving them into a framework for creative adaptation and mathematical dramatization.

🔧 Constructing a 4×4 Magic Square with S = 40 Using Kuttaka

We begin with Narāyaṇa’s Diophantine setup:

$$S=4a+30d\Rightarrow 40=4a+30dS\; =\; 4a\; +\; 30d\; \backslash Rightarrow\; 40\; =\; 4a\; +\; 30d$$

This equation yields infinite integral solutions for $(a,d)(a,\; d)$, such as:

• $(a,d)=(-5,2)(a,\; d)\; =\; (-5,\; 2)$

• $(a,d)=(10,0)(a,\; d)\; =\; (10,\; 0)$

• $(a,d)=(25,-2)(a,\; d)\; =\; (25,\; -2)$

Each pair defines a linear sequence:

$$a,a+d,a+2d,\dots ,a+15da,\; a\; +\; d,\; a\; +\; 2d,\; \backslash dots,\; a\; +\; 15d$$

which populates the square. The arrangement (via Turagagati or other methods) ensures the magic sum $S=40S\; =\; 40$ across rows, columns, and diagonals.

🌀 Toroidal Mapping and Pan-Diagonal Properties

Mapping the square onto a torus—where opposite edges are identified—reveals deeper symmetries:

🧩 Property 1: 2×2 Sub-Squares Sum to 34

In the classic square:

Code

 1  12 13  8
15   6  3 10
 4   9 16  5
14   7  2 11

Any 2×2 block (even wrapping around edges) sums to 34. This reflects local balance—ideal for staging as rhythmic or musical motifs.

🔁 Property 2: Diagonal Pairs Sum to 17

Across the torus:

• $1+16=171\; +\; 16\; =\; 17$

• $6+11=176\; +\; 11\; =\; 17$

• $15+2=1715\; +\; 2\; =\; 17$

• $4+13=174\; +\; 13\; =\; 17$

• etc.

This reveals a hidden duality—each number paired with its complement. Dramatically, this could symbolize opposing forces, mirrored characters, or ritual balance.

🧭 Property 3: Neighbours of 16

In the toroidal grid, the “neighbours” of 16 are:

• 3 (left)

• 5 (right)

• 2 (above)

• 9 (below)

These are always the same set—2, 3, 5, 9—regardless of rotation. This invariant neighborhood could be visualized as a protective ring or symbolic guardianship.

🎭 Creative Adaptation Potential

• Visual Treatment: Use toroidal mapping to design circular stage layouts or rotating grids.

• Musical Phrasing: Map 2×2 blocks to rhythmic cycles; diagonal pairs to harmonic intervals.

• Character Dynamics: Let each number represent a persona—its neighbours define relationships, its diagonal complement a hidden twin.

• Symbolic Encoding: Use the (a, d) sequence as a metaphor for destiny and transformation.

Would you like to construct a few squares with different $(a,d)(a,\; d)$ pairs and explore how their internal symmetries shift? Or perhaps map these properties to dramatic motifs—light, sound, movement? I’d love to co-create that with you.