Maths Magic in Jainism

Note: A magic square is a ‘n x n’ square grid (where ’n’ is the number of cells on each side) filled with distinct positive integers in the range 1, 2…, n2 such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the ‘Magic Constant’ or ‘Magic Sum’ of the magic square.

1. Numbers used in the Magic Square
2. Number of ways we get the ‘Magic Constant’
3. The Yantra Itself

Let’s discuss them all.

Numbers used in the Magic Square

Per definition, a normal magic square of ‘n x n’ will have numbers with the below specifications: –

1. Numbers are in sequence
2. Numbers starting from 1
3. Up to n2

Figure (A) shows the ‘Uvasaggaharam Magic Square’ known as non-normal magic square.

On initial look it seems it has some random numbers.

1. Numbers are out of sequence
2. Numbers are starting from 0
3. Up to 3n2–1

Here are the numbers in ascending order — [0,5,10,15,20,26,27,28,31,32,33,36,37,38,41,42,43,46,47,48,54,59,64,69,74] which doesn’t give any pattern. But when we take the difference of the consecutive numbers (m — m-1) we have beautiful palindrome number sequence — [5,5,5,5,6,1,1,3,1,1,3,1,1,3,1,1,3,1,1,6,5,5,5,5]. Plotting a line graph in Figure (B) which is symmetric on ‘Y’ axis, meaning there is a sequence/pattern followed.

Trying a different approach, write all the number from 0 to 74 in group of 5 and highlighting the magic numbers give the ‘Glide Reflection Symmetric’ Figure ©

Number of ways we get the ‘Magic Constant’

Per definition, a regular 5×5 will have a magic constant of 65, but this ‘Uvasaggaharam Magic Square’ has it as 185. Since the ‘Uvasaggaharam Sotra’ has 185 characters the Magic Square is constructed for the constant of 185. A normal 5×5 magic square has 12 different quintets to get the magic constant as 65. Let’s see how many ways we get 185.

Considering the rows

Figure (1) to (5) shows all the individual rows adds up to 185

Considering the Columns

Figure (6) to (10) shows all the individual rows adds up to 185

Considering the Diagonals

Diagonals are further divided into 2 parts:

Normal Diagonals

Figure (11) and (12) shows the 2 normal diagonals adds up to 185

Broken Diagonals

A Broken diagonal is a set of n cells forming two parallel diagonal lines in the square. Alternatively, these two lines can be thought of as wrapping around the boundaries of the square to form a single sequence. Figures (13) to (16) & (17) to (20) corresponds to the 2 sets of Broken Diagonal making this Magic Square as ‘Pandiagonal Magic Square’

Subset of 3×3

Every 5×5 square has 9 different 3×3 internal squares as show below in figures ‘a’ to ‘i’.

And each of these 3×3 squares have 2 set of quintets as below.

Thus, we have total of 18 new quintets with the sum of 185 as shown in figures (21) to (38)

Special Ones

In the above all the 3×3 the 2 quintets are formed as

1. 4 Edges and the Centre of the 3×3
2. 4 Corners and the Centre of the 3×3

Applying the same logic to 5×5 square we get 2 more quintets of 185, Figures (39) and (40)

In this way we see a total of 40 Quintets formed in this 5×5 Magic Square.

Bonus

The Scriptures of this Yantra were having a very good hold in Maths and they hide a group of 9 digits to give a sum of 185–4 Broken Diagonals and a Centre. Figure (41)

Acknowledgement:  https://medium.com/@connect_26403/maths-magic-in-jainism-c83579c45b68

Author: Bhavin Mehta