Well done Geoffrey, University College School, London for this excellent solution.
This problem is actually remarkably straightforward. A little bit of algebra is all that is needed.
Let x and y each denote 1 of the 2 numbers to be multiplied together.
The layout of the grids in the question then corresponds to:
|x||( 10 - x )|
|y||( 10 - y )|
The 4 ways of calculating the left hand digit are expressed algebraically as follows:
All 4 ways are algebraically equivalent, so this is why they give the same answer.
To explain why the Vedic Sutra works, we run through the whole method algebraically.
The right hand digit is ( 10 - x ) * ( 10 - y ).
The left hand digit is ( x + y - 10 ). We should remember that this is the tens digit and multiply by 10 before summing both sides.
10 * ( x + y - 10 ) + ( 10 - x ) * ( 10 - y )
= 10x + 10y - 100 + 100 - 10x - 10y + xy
Using the Vedic Sutra, we get the result xy, which is indeed the product of x and y.