**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 C_{1,}

_{ }C_{1 }= **0588235294117647** * **123456787654321**

——————————————————–

** 072621639796659404502541902687 **

——————————————————–

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

2) Split C_{1} 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:

D_{1} = 4117647058823529

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

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

4) Now multiply D_{1}with a palindrome number ‘B’, and repeat the steps above. again, interactively.

I.e. eg : C_{2 }= D_{1}* B = **508351478576615831517793318809**

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

*508351478576615

831517793318809+

Add the two to get D_{2} = **8823529411764705.**

Note that D_{2} again is similar to A, except there is a shift in the sequence!

Repeat this procedure continuously, to get, in turn,

D_{3}, …D_{4, }…D_{16}.

These D_{i ,} 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