# Using Duplexes

In our post about combining partial sums in the previous issue of the MATH-Inic Newsletter, we showed how to evaluate this: **16/56/97/108/78/36/9.**

This is the final step in answering this question from the Intermediate age group of the 4^{th} MATH-Inic Vedic Mathematics National Challenge held last April 20, 2024: What is the square of **4763 **?

This question is easily answered using duplexes.

The duplex of a single digit is its square while the duplex of a two digit number is twice the product of the two digits.

The general rules for longer numbers are:

- For those with even number of digits, the duplex is twice the sum of the products of the symmetrically placed digits about the center.

- For those with odd number of digits, the duplex is twice the sum of the products of symmetrically placed digits about the center plus the square of the middle digit.

The Duplex, D, of the following numbers are:

D(a) = a^{2}

D(ab) = 2ab

D(abc) = 2ac + b^{2}

D(abcd) = 2ad + 2bc

D(abcde) = 2(ae + bd) + c^{2}

The square of a number is the sum of its duplexes.

4763^{2} = D(4) + D(47) + D(476) + D(4763) + D(763) + D(63) + D(3)

= 16/56/97/108/78/36/9

= **22,686,169**