Deriving Lakhan function

                                         courtesy: imageshack

Its acceptable to assume that Indians (if u aren't on Mars mission since 1989 or born recently) are well aware of Mr. Lakhan and his equally famous Equation - One Two ka Four...Four Two ka One.

I saw a post recently and decided to apply some mathe-magic to derive this function (if its not one of Bollywood's 'suspension-of-disbelief'). So, cross your fingers and scroll below...

To Derive: 
a) One and Two with a Mathematical operator should result into Four
b) Four and Two with Same Mathematical operator should result into One

Function:
a) f(x,y) = 4 when x=1, y=2
b) f(x,y) = 1 when x=4, y=2

say, f(x,y) = mx OP ny (also ny OP mx, in case of / and -) where { m,n ∈ R } and OP = { +, -, *, / }   [For brevity, we have selected the four-most operators. You are free to apply others and share the Result]

Thus,
a) 1m OP 2n = 4
b) 4m OP 2n = 1

CASE: OP = '+'
a) 1m + 2n = 4
b) 4m + 2n = 1

a) - b) : -3m = 3
           : m = -1
Thus, n = 2.5

f(x,y) = -1x + 2.5y                       Result 1

Wow, Strike in first try.............lets move on with other operators


CASE: OP = '-'
a) 1m - 2n = 4
b) 4m - 2n = 1

again, a) - b) : -3m = 3
                     : m = -1
Thus, n = -2.5

f(x,y) = -1x + 2.5y                       Result 1 verified (reversing the operands will result into same equation)


Lets see if we can get something simpler...

CASE: OP = '*'
a) 1m * 2n = 4
b) 4m * 2n = 1

a) / b) : 1/4 = 4    (clearly not possible till 10/26/2012)

CASE: OP = '/'
a) 1m / 2n = 4
b) 4m / 2n = 1

a) / b) : 1/4 = 4    (again not possible)

CASE: OP = '/' (reverse operands)
a) 2n / 1m = 4
b) 2n / 4m = 1

or, n = 2m
i.e. The equations are True for any n = 2m.

Say m = 1, then n= 2
f(x,y) =  2y / x                        Result 2


Lakhan Function

               f(x,y) = (5y-2x)/2 or 2y/x

So, with little bit of Maths and free time we have Proven the Genuinity of Mr. Lakhan. 
Hat's off to Anil and Subhash ghai...kudos Bollywood...