Little Einstein writes the numbers 2 , 3 , … N on a blackboard. He chooses any two of them, say X and Y, scratches both of them and replaces them by the single
number T given by the relation :
T = XY – X – Y + 2
He repeats this process until there is only one number left on the board. He notes down this number and starts again from the beginning. To his utter bafflement, he ends up with the same 'magic' number, irrespective of how he does it.
What is this magic number?
21Here we look for what does not change.(method of invariants). We replace x and y by xy-x-y+2= (x-1)(y-1)+1
So, if we subtract 1, from each number and multiply them , the product will be constant in any step.
Initially the value of the product was (n-1)!
So when we are left with only two numbers call a and b, we must have (a-1)(b-1) = (n-1)!
In the last step we replace a,b by (a-1)(b-1) + 1 = (n-1)! + 1
Hence we will always be left with (n-1)! + 1
