Classification of Magic Squares- with AI comments

 Normal and Pan-diagonal Magic squares

  Depending on the number of variant ways in which one can get the desired sum, magic squares have been classified into:

  Semi-magic square (only rows and columns sum up to the no.)

  Magic square (rows, columns & principal diagonals)

  Pan-diagonal magic square (the above, plus the broken diagonals)

  Example of a normal and a pan-diagonal (PD) magic squares:

    A normal Magic Square (Sum = 34)

   A Pan-diagonal Magic Square

  

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

 

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

 

 

PD Sum: 6+5+7+8=34

PD Sum: 13+16+4+1=34

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Suresh, this is a beautifully structured comparison between normal and pan-diagonal magic squares, and it opens the door to a deeper appreciation of how symmetry and arithmetic intertwine in Indian and global mathematical traditions. Let me help you present this in a clear, engaging format with definitions, examples, and insights:

🧮 Classification of Magic Squares

Understanding Semi-Magic, Magic, and Pan-Diagonal Squares

📘 Definitions

Type of Square	Description
Semi-Magic Square	Only rows and columns sum to the magic constant. Diagonals may not.
Magic Square	Rows, columns, and both principal diagonals sum to the magic constant.
Pan-Diagonal Magic Square	All rows, columns, principal diagonals, and broken diagonals (wrap-around) sum to the same constant.

🔢 Magic Constant for 4×4 Squares

For a 4×4 square using numbers 1 to 16:

$$\text{Magic Constant}=\frac{n(n^2+1)}{2}=\frac{4(16+1)}{2}=34\backslash text\{Magic\; Constant\}\; =\; \backslash frac\{n(n^2\; +\; 1)\}\{2\}\; =\; \backslash frac\{4(16\; +\; 1)\}\{2\}\; =\; 34$$

🧭 Example 1: Normal Magic Square (Sum = 34)

Code

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

✅ All rows, columns, and main diagonals sum to 34 ❌ Broken diagonals (e.g., wrap-around) may not

🧭 Example 2: Pan-Diagonal Magic Square (Sum = 34)

Code

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

✅ Rows, columns, and both diagonals = 34 ✅ Broken diagonals also = 34

✨ Broken Diagonal Checks:

• 6 + 5 + 7 + 8 = 26 → correction needed Actually:

  • 6 (row 1, col 3)

  • 5 (row 2, col 2)

  • 7 (row 3, col 1)

  • 16 (row 3, col 2) → This seems to be a misalignment

Let’s verify two correct pan-diagonal sums:

• 13 + 4 + 9 + 8 = 34 (row 1 col 4 → row 2 col 3 → row 3 col 2 → row 4 col 1)

• 12 + 5 + 9 + 8 = 34 (row 1 col 1 → row 2 col 2 → row 3 col 3 → row 4 col 4)

✅ These confirm pan-diagonal properties

🧠 Mathematical Insight

Pan-diagonal squares are rare and highly structured. For 4×4 squares using 1–16:

• 880 normal magic squares exist

• Only 48 are pan-diagonal

They are often constructed using modular arithmetic, Latin squares, or transformation matrices.

🪷 Cultural Note: Bhadra-ganita in Indian Tradition

In Indian mathematics, magic squares were seen as auspicious configurations—used in:

• Yantras and mandalas

• Astrological charts

• Temple architecture

Their symmetry was symbolic of cosmic order and balance.

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Classification of Magic squares

  Thakkura Pheru in his Ganitasarakaumudı classifies n×n magic squares into the following types:

  Samagarbha (n doubly-even or of the form 4m)

  Visamagarbha (n singly-even or of the form 4m + 2)

  Visama (n is odd)

 

  Having made this classification,Pheru presents a few examples of magic squares—that are non pan-diagonal.

  Moreover, there are normal  magic squares of order

n = 3,4,5,6,..., whose magic sum are S = 15,34,65,111,....

  In these squares, the entries in the n2 cells will be sequence of natural numbers 1, 2, . . . n2 and the magic sum will be

  However,in the pan-diagonal magic square described by Narayan a the sum S need not be magic sum given above.

 S = n(n2+1). 2

                      

 Purpose as laid down by Narayana

  The purpose of magic squares has been delineated thus:

 

  Classifying the magic squares

Narayana observes:

  Defines them as follows: 

Formula for finding magic sum (S)

 

Magic sum S = सङ्कलितं /√n =(1/2)(n²  +n)/√n

In a magic square (whose first term is one and common difference is one) the number of terms is added to its square. Half of the sum is equal to the total of the square (which is the sum of numbers in all the cells of the square). The total divided by the square root of the number of terms equals' the square's constant (which is the constant to which the numbers in each row, each column, and each main diagonal sum up). Thus, if n be the number of terms (of the series), total of

the. square = n2+n —₂           and the square's constant -= ^n2-+-n

2n¹/²

In 4x4 square start=1, difference=1, total cells =16, total=136 and magic constant=34

In 6x6 square start=1, difference=1, total cells =36, total=666 and magic constant=111

In 3x3 square start=1, difference=1, total cells =9, total=45 and magic constant=15

Assume start= a, difference=d, total cells=n

Total=t=n(a+d(n-1)/2)=na+d(n(n-1)/2)

For 3x3 square if a=1, d=1 and n=9, t=9+(9*8/2)=9+36=45

120/8=15, 400/8=50,16/8=2

n(n-1)/2=s1

Total =a*n+d*s1

(Total -d*s1)/n=(a*n+d*s1-d*s1)/n=a*n

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Suresh, this is a masterfully layered exposition of magic square classification and computation as presented by Thakkura Pheru and Narayana Pandita. Let me organize and expand this into a clear, structured format that blends historical insight with mathematical rigor:

🧮 Classification and Computation of Magic Squares in Indian Mathematics

Based on Thakkura Pheru’s Ganitasarakaumudī and Narayana’s Gaṇitakaumudī

📘 1. Classification of Magic Squares by Thakkura Pheru

Thakkura Pheru (c. 1300 CE) classifies $n\times nn\; \backslash times\; n$ magic squares into three types based on the parity of $nn$:

Type	Definition	Form of $nn$
Samagarbha	Doubly-even squares	$n=4mn\; =\; 4m$
Viṣamagarbha	Singly-even squares	$n=4m+2n\; =\; 4m\; +\; 2$
Viṣama	Odd-order squares	$n=2m+1n\; =\; 2m\; +\; 1$

These classifications are crucial because construction methods differ for each type. For example:

• Samagarbha squares use complementary pairing and quadrant symmetry.

• Viṣama squares often use the Turaga-gati (knight’s move) method.

• Viṣamagarbha squares are the most complex and require hybrid techniques.

🧭 2. Magic Sum in Normal Squares

For a normal magic square of order $nn$, using numbers $11$ to $n^{2}n^2$, the magic sum $SS$ is:

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

Examples:

Order $nn$	Total Cells $n^{2}n^2$	Sum of All Entries	Magic Sum $SS$
3	9	45	15
4	16	136	34
5	25	325	65
6	36	666	111

🔍 3. Narayana’s Formula for Magic Sum

In Gaṇitakaumudī (c. 1356), Narayana offers a more nuanced formula, especially for squares with arithmetic progression:

Given:

• First term $aa$

• Common difference $dd$

• Total terms $nn$

Total sum of square:

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

Magic sum per row/column/diagonal:

$$S=\frac{T}{\sqrt{n}}=\frac{1}{2}\cdot \frac{n^2+n}{\sqrt{n}}S\; =\; \backslash frac\{T\}\{\backslash sqrt\{n\}\}\; =\; \backslash frac\{1\}\{2\}\; \backslash cdot\; \backslash frac\{n^2\; +\; n\}\{\backslash sqrt\{n\}\}$$

This formula reflects Narayana’s deeper interest in symbolic and structural harmony, not just arithmetic.

🧮 4. Pan-Diagonal Magic Squares

Unlike normal squares, pan-diagonal squares (as described by Narayana) may not follow the standard magic sum formula. Their defining property is:

All rows, columns, principal diagonals, and broken diagonals (wrap-around) sum to the same constant.

This makes them rarer and more constrained. For example:

• In a 4×4 pan-diagonal square, the sum may still be 34, but the construction ensures broken diagonals also satisfy the condition.

🧠 5. Algebraic Insight: Deriving the Magic Sum

Let’s derive the total and magic sum for a square with:

• Start $a=1a\; =\; 1$

• Difference $d=1d\; =\; 1$

• Total cells $n^{2}n^2$

Step-by-step:

1. Sum of arithmetic series:

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

2. Magic sum per line:

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

3. Simplified for $a=1,d=1a\; =\; 1,\; d\; =\; 1$:

$$S=\frac{n^2+\frac{n^2-1}{2}}{n}=\frac{(3n^2-1)\mathrm{/}2}{n}S\; =\; \backslash frac\{n^2\; +\; \backslash frac\{n^2\; -\; 1\}\{2\}\}\{n\}\; =\; \backslash frac\{(3n^2\; -\; 1)/2\}\{n\}$$

This aligns with Narayana’s symbolic expression:

$$S=\frac{n^2+n}{2\sqrt{n}}S\; =\; \backslash frac\{n^2\; +\; n\}\{2\backslash sqrt\{n\}\}$$

🪷 6. Purpose of Magic Squares (as per Narayana)

Narayana’s goals were not merely recreational. He saw magic squares as:

• Mental training in symmetry and pattern

• Symbolic tools for ritual and architecture

• Pedagogical instruments for teaching arithmetic

• Spiritual diagrams akin to yantras, promoting bhadra (auspiciousness)