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.

The 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) = 3² = 9 D(345) = 4² + 2 ( 3 x 5) = 16 + 30 = 46 D(abcde) = c² + 2 x [(ae) + (bd)] D(76543) = 5² + 2 x [(7 x 3) + (6 x 4)] = 25 + 2(45) = 115 The square of a number is the sum of its duplexes. In our example, 364² = D(3) + D(36) + D(364) + D(64) + D(4)

          D(3) = 3² = 9

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

D(364) = 2(3 x 4) + 6² = 24 + 36 = 60

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

D(4) = 4² = 16

There are others ways to get the sum of the duplexes, aside from the method shown in the figure.   This is another one: 364² = 9/36/60/48/16. To evaluate this, we must start from the right and follow these steps:

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

The final answer, 132,496  is obtained. This 3-line solution is recommended for squaring larger numbers. 4,357²  = D(4) + D(43) + D(435) + D(4357) + D(357) + D(57) + D(7) D(4) = 4² = 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) + 3² = 40 + 9 = 49

1 6  4  9

               2  4 D(4357) = 2[(4 x 7) + (3 x 5)] = 2(28 + 15) = 86 D(357) = 2(3 x 7) + 5² = 42 + 25 = 67 D(57) = 2(5 x 7) = 70 D(7) = 7² = 49

1 6  4  9  6  7  0  9

               2  4  8  6  7  4___    4357² =1 8, 9  8  3, 4 4  9  Now, can you see how the squaring of this polynomial is done in one line? (3x² + 4x + 1)² = 9x⁴ + 24x³+ 22x² + 8x + 1