July 2023

Mathematical Matrix


16 X 16 Prime Reciprocal Magic Square

A 18 * 18 Magic square has long been known, we present here a procedure for generating a 16 * 16 Magic square may be for the very first time, using a very interesting method.

Let us start with 2 numbers A & B.

‘A’ is the reciprocal of prime number 17

A = 1/17 = 0.0588235294117647

Let us ignore the decimal point and write A as:  0588235294117647

B = 123456787654321 is a 15 digit palindrome number (Considering the digits on either side of ‘ 8 ‘).

We now adopt the following procedure:

1) Multiply A with B to get C1,

       C1 = 0588235294117647 * 123456787654321




Which is 30 digits long (including the leading ‘0‘)

2) Split C1 equally into 2 parts, writing it down as,



Note that there is an offset or indent in the way the numbers have been written one above the other.

3) Add the two rows as written to get:

D1 = 4117647058823529

Note that this result is similar to the number A, except that the last 7 digits in A are now the 1st seven

and the 1st 9 digits in A are now the last nine!

4) Now multiply D1with a palindrome number ‘B’, and repeat the steps above. again, interactively.

I.e.  eg :  C2 = D1* B = 508351478576615831517793318809

Split these 30 digits again into two parts (as in step 2) to get



Add the two to get D2 = 8823529411764705.

Note that D2 again is similar to A, except there is a shift in the sequence!

Repeat this procedure continuously, to get, in turn,

D3, …D4, …D16.

These Di ,  i = 1 to 16, when written top to bottom in sequence, form a 16*16 grid.

The sum of this 16*16 square along any column, row or diagonal, each yield the number 72!

This is a “Magic Square”

Email: Subramani.k@iiitb.ac.in


The International Institute of Information Technology Bangalore is a technical and research university in Bangalore, India. The Institute is a registered not-for-profit society funded jointly by the Government of Karnataka and the IT industry under a public-private partnership model.

For more details: Visit www.iiitb.ac.in

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