  July 2023

# Mathematical Matrix

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## 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.

‘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

——————————————————–

072621639796659404502541902687

——————————————————–

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

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

*072621639796659

404502541902687+

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

*508351478576615

831517793318809+

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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