Duplex

The Duplex of a number, also known as the Dvandva Yoga of a number, helps us to calculate Squares and Square Roots.

General Method of finding the Duplex of a number

To find the Duplex of a number you have to keep finding double the product of digits which are equidistant until you reach the centre of the number. If, when you reach the centre there is only one digit, take its square, then.....add all the products together and the result is the Duplex of that number. So, by doubling the first and last digit of the number, then the second and the last-but-one, then the third and the last-but-two followed by the fourth and the last-but-three and so on. Eventually, you will arrive at the centre and adding up all of the products will give the Duplex of the number.

Single digit number

For a single digit number, "a", the Duplex, D(a), is simply the square of the number. Thus D(a) = a²

• Duplex of 2, D(2) = 2² = 4
• Duplex of 6, D(6) = 6² = 36
• Duplex of 8, D(8) = 8² = 64 etc.

Two Digit Number

For a two digit number "ab", the Duplex, D(ab), is twice the product of these two digits. Thus D(ab) = 2ab

• Duplex of 32, D(32) = 2 x (3 x 2) = 12
• Duplex of 57, D(57) = 2 x (5 x 7) = 70
• Duplex of 89, D(89) = 2 x (8 x 9) = 144 etc.

Three Digit Number

For a three digit number "abc", the Duplex, D(abc), is obtained by adding twice the cross product of the 1st & 3rd digits and the square of the middle digit. Thus D(abc) = 2ac + b²

• Duplex of 213, D(213) = 2 x (2 x 3) + 1² = 12 + 1 = 13
• Duplex of 456, D(456) = 2 x (4 x 6) + 5² = 48 + 5 = 53
• Duplex of 879, D(879) = 2 x (8 x 9) + 7² = 144 + 49 = 193 etc.

Four Digit Number

For a four digit number "abcd" the Duplex, D(abcd), is obtained by adding twice the cross product of the 1st & 4th digits and twice the cross product of the 2nd & 3rd digits. Thus D(abcd) = 2ad + 2bc

• Duplex of 4632, D(4632) = 2 x (4 x 2) + 2 x (6 x 3) = 16 + 36 = 52
• Duplex of 7049, D(7049) = 2 x (7 x 9) + 2 x (0 x 4) = 126 + 0 = 126
• Duplex of 9789, D(9789) = 2 x (9 x 9) + 2 x (7 x 8) = - 162 + -112 = -274 etc.

Five Digit Number

For a five digit number "abcde", the Duplex, D(abcde), is obtained by adding together i) twice the product of the 1st and 5th digit, ii) twice the product of 2nd and 4th digits and, finally.... iii) the square of the third digit. Thus D(abcde) = 2ae + 2bd + c².

• Duplex of 42361, D(42361) = 2 x (4 x 1) + 2 x (2 x 6) + 3² = 8 + 24 + 9 = 41
• Duplex of 75289, D(75289) = 2 x (7 x 9) + 2 x (5 x 8) + 2² = 126 + 80 + 4 = 210
• Duplex of 52461, D(52461) = 2 x (5 x 1) + 2 x (2 x 6) + 4² = 10 + 24 + 16 = 50 etc.

In this way, following the Duplex of any number can be found directly.

**Please note that:

• The Duplex of a number is the same when the number is reversed from front to back
• The Duplex of an n-digit number is always even when n is even
• The Duplex of an n-digit number is always odd when both n and the middle digit of the number are odd