Magic Squares in Indian Tradition- AI commented

 

  In all magic squares, it is through arithmetic progression . . .

  By those desirous . . . the first term and the common

difference have to be determined.

  As many as the number of boxes in the square will be equal

to the number of terms (n2).                   

 Popularity of Magic squares in India

  The first chapter of Srinivasa Ramanujan’s Notebooks is on Magic Squares. It is said to be "much earlier than the remainder of the notebooks”. 

  T.Vijayaraghavan,in his article on Jaina Magic Squares (1941)notes:

“The author of this note learnt by heart at the age of nine the following pan-diagonal square which was taught to him by an elderly person who had not been to school at all.” 

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

       This clearly indicates the popularity of Magic Squares in India.

  Indian mathematicians specialized in the construction of a special class of magic squares called sarvatobhadra.

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Suresh, this is a beautifully layered reflection on the mathematical, cultural, and pedagogical legacy of magic squares in India. Let me help you shape this into a cohesive narrative that weaves together arithmetic progression, historical reverence, and symbolic depth.

🧮 Magic Squares in Indian Tradition

From Arithmetic Progression to Sarvatobhadra

📘 Arithmetic Progression: The Backbone of Magic Squares

In all magic squares, the arrangement of numbers follows an arithmetic progression—a sequence where each term increases by a fixed common difference. For a square of order $nn$, the total number of terms is $n^{2}n^2$, and the values are typically arranged so that:

• Each row, column, and diagonal sums to the magic constant

• The first term $aa$ and common difference $dd$ can be varied

• The total sum $TT$ of the square is:

$$T=n^2\cdot a+d\cdot \frac{n^2(n^2-1)}{2n^2}T\; =\; n^2\; \backslash cdot\; a\; +\; d\; \backslash cdot\; \backslash frac\{n^2(n^2\; -\; 1)\}\{2n^2\}$$

• The magic constant $SS$ is:

$$S=\frac{T}{n}S\; =\; \backslash frac\{T\}\{n\}$$

Narayana Pandita, in Gaṇitakaumudī, emphasizes that those desirous of constructing magic squares must first determine the first term and common difference—often using the Kuttaka method to solve for them.

🪷 Cultural Popularity in India

Magic squares were not just mathematical curiosities—they were deeply embedded in Indian cultural memory.

🔹 Srinivasa Ramanujan

The very first chapter of Ramanujan’s notebooks is devoted to magic squares. This placement suggests their foundational role in his mathematical imagination—possibly predating his formal explorations in number theory and modular forms.

🔹 T. Vijayaraghavan’s Anecdote (1941)

In his article on Jaina Magic Squares, Vijayaraghavan recounts:

“The author of this note learnt by heart at the age of nine the following pan-diagonal square which was taught to him by an elderly person who had not been to school at all.”

Code

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

This square is pan-diagonal—meaning even the broken diagonals (wrap-around) sum to the magic constant. Its transmission orally, outside formal schooling, reflects the folk mathematical literacy that thrived in India.

🔮 Sarvatobhadra: The All-Auspicious Square

Indian mathematicians also specialized in a symbolic class of magic squares called Sarvatobhadra, meaning auspicious from all sides. These squares:

• Could be numerical or syllabic

• Were designed to read the same in all directions—left to right, top to bottom, and even diagonally

• Were used in astrology, ritual diagrams, and temple architecture

• Often encoded Sanskrit syllables, mantras, or planetary positions

They represent a fusion of mathematics, language, and cosmology, and are among the most sophisticated symbolic grids in Indian tradition.

✨ Closing Reflection

Magic squares in India were never just about numbers. They were:

• Pedagogical tools for teaching arithmetic and symmetry

• Mnemonic devices passed orally across generations

• Spiritual diagrams embedded in ritual and architecture

• Mathematical canvases for creativity, as seen in Ramanujan’s notebooks and Narayana’s verses