# MSC #26: Duplex

A question in the Junior Level of the 2^{nd} International Vedic Mathematics Olympiad requires the answer to the square of 2022. Because the figure only involves 2s and 0, this is easily solved even when using conventional multiplication.

But squaring numbers such as **4,357** would not be easy without using duplexes, which is the subject of the **26 ^{th}** of our

**30**MATH-Inic Specials for Christmas series.

**Duplex**is an important device in squaring numbers and polynomials, extracting square roots and solving quadratic equations. Its principles can be extended into triplex and beyond. The duplex of a number with an

**even**number of digits is twice the sum of the products of the symmetrically placed digits.

**D(ab) = 2ab**

**D(46) = 2 (4 x 6) = 48**

**D(abcd) = 2 x (ad + bc)**

**D(2345) = 2 x [(2 x 5) + (3 x 4)] = 44**The duplex of a number with an

**odd**number of digits is square of the middle digit plus twice the sum of the products of to the symmetrically placed digits.

**D(3) = 3**

^{2}= 9**D(345) = 4**

^{2}+ 2 ( 3 x 5) = 16 + 30 = 46**D(abcde) = c**

^{2}+ 2 x [(ae) + (bd)]**D(76543) = 5**The square of a number is the sum of its duplexes. In our example,

^{2}+ 2 x [(7 x 3) + (6 x 4)] = 25 + 2(45) = 115**364**

^{2}= D(3) + D(36) + D(364) + D(64) + D(4)** D(3) = 3 ^{2 }= 9**

**D(36) = 2(3 x 6) = 36**

**D(364) = 2(3 x 4) + 6 ^{2} = 24 + 36 = 60**

**D(64) = 2(6 x 4) = 48**

**D(4) = 4 ^{2} = 16**

**364**To evaluate this, we must start from the right and follow these steps:

^{2 }= 9/36/60/48/16.1) Bring down **6**

2) Add the 1 of 16 to the 8 of 48 to get **9** and place it to the left of **6** in the answer.

3) Repeat step 2 until the leftmost duplex

a) 4 + 0 = **4**

b) 6 + 6 = 12; write down **2** and “carry” 1

c) 3 + 9 + the carry digit 1 = **13**

^{2}= D(4) + D(43) + D(435) + D(4357) + D(357) + D(57) + D(7) D(4) = 4

^{2}=

**16**Write down

**16**

**1 6**D(43) = 2(4 x 3) =

**24**Write down

**24**as shown below 1 6

**4**

**2**D(435) = 2(4 x 5) +

**3**

^{2}

**=**40 + 9 =

**49**

1 6 4 **9**

**4**D(4357) = 2[(4 x 7) + (3 x 5)] = 2(28 + 15) =

**86**D(357) = 2(3 x 7) + 5

^{2}= 42 + 25 =

**67**D(57) = 2(5 x 7) =

**70**D(7) = 7

^{2}=

**49**

1 6 4 9 **6 7 0 9**

__2 4__

**8 6 7 4___****4357**

^{2}=

**1 8, 9 8 3, 4 4 9**

**Now, can you see how the squaring of this polynomial is done in one line?**

**(3x**

^{2}+ 4x + 1)^{2}= 9x^{4}+ 24x^{3}+ 22x^{2}+ 8x + 1