Yes, but if you think it is very far away from mathematics, you're wrong.
Many interesting Combinatorial problems requires computational power for solving. But does that mean computers can reduce the problems into pure bashing? Nope! As you can see there, the backtracking solution can only solve upto the 10 by 10 chessboard whereas the one supplemented by enough research can solve upto a 174 by 174 chessboard.
I found another solution (much like this one) on another forum: https://brilliant.org/problems/too-many … hessboard/
You will need a computer for this: http://codegolf.stackexchange.com/quest … hess-board
One solution is a backtrack solution. Another guy (elsewhere) gave me a gf solution which I did not really understand that well.
We are using strong induction here.
First we show that it is true for some of the beginning values by computation.
Then, we assume that for some value n, all of the preceeding values show the proposed property.
We now need to show that n also shows the same property, to complete the induction.
Note that f(n) = f(n-2^0) + f(n-2^1) + ... + f(n-2^k) + ...
We're interested in if any of these are odd. By the induction hypothesis, if there is such a term, it should be of the form 2^j - 1. This is because we've chosen to accept that only the output of such numbers are odd.
So, n - 2^k = 2^j - 1