Vedic Sutra #1:     By one more than the one before.

We want to square 65. That is, calculate 65².

1. We first note that the number ends in "5" ... that's important!
2. We square this second digit. That gives 5*5 = 25 ... which is how our answer ends!
3. Then we pick the first digit. That's "6".
4. We multiply "6" by "7" giving 42 ... which is how our answer begins!
   Why 7? That's the "one more than the one before", since 7 is one more than 6, eh?
5. The answer is 4225.

Note that 65 = 10*6 + 5 and, in general , we consider a number 10x + 5.
Then (10x + 5)² = 100 x² + 100x + 5² = 100x(x+1) +25
and we see the magic "one more than the one before", namely: x(x+1).

In particular, 65² = 100[6(6+1)] + 25 = 100(42)+25 = 4225.

It's that "100" out front that moves 42 into first-two-digits position.