Dwandwa Yoga
The Dwandwa Yoga or 'duplex combination' can be used for general purpose squaring.
To proceed further we need to know the Dwandwa of certain numbers.
D( a ) = a2
D( ab ) = 2ab
D( abc ) = 2ac + b2
D( abcd ) = 2ad + 2bc
D( abcde ) = 2ae + 2bd + c2
D( abcdef ) = 2af + 2be + 2cd and so on....
As we can see above, D of any number is the sum of square of the middle number and two times the product of the other pairs.
Square of a number is given by
( ab )2 = D( a ) | D( ab ) | D( b )
( abc )2 = D( a ) | D( ab ) | D(abc) | D( bc ) | D( c )
( abcd )2 = D(a) | D(ab) | D(abc) | D(abcd) | D(bcd) | D(bc) | D (c)
Example :
(23)2 = (ab)2 = D(a) | D(ab) | D(b)
= 4 | 12 | 9
Since Dwanada must have only one digit, we carry over '1' of '12' to LHS.
Therefore it becomes 4 |1 2| 9
Hence the answer is 529
Example 2:
(527)2 = ( abc )2
= D( a ) | D( ab ) | D(abc) | D( bc ) | D( c )
= 25 | 20 | 74 | 28 | 49
= 25 |2 0 |7 4 |2 8 |4 9
= 277729