N-Puzzle

This is a beautiful puzzle called the N-Puzzle- it is short , simple but in no way sweet. We took it to Lucknow for the IYMC competition for a project display competition.

Ok, so all of us may have seen a sort of game that looks like the image below and played with it when we were kids :-

But here's what a curious person named Noyes Chapman did. He interchanged the position of 14 and 15. Now the basic puzzle looked as follows:-

Now it was given to be solved , that is, the aim was to get the above arrangement from a random arrangement.
And surprisingly, even though it looks easy at first, it was found to be insolvable.

This puzzle is also known by Fifteen Puzzle, Gem Puzzle, Boss Puzzle, Game of Fifteen or Mystic Square.

To prove- Fifteen Square is Unsolvable

Proof:-(Courtesy:Wikipedia. en.wikipedia.org)
The proof involves noting that there are two distinct sets of positions which can be assembled from the pieces with different parity, and there is no way of moving between them using the allowed moves, as they preserve parity. The parity in this context (the invariant) is the parity (odd or even) of the number of pairs of pieces in reverse order plus the row number of the empty square. For the order of the 15 pieces consider line 2 after line 1, etc., like words on a page. Thus an even permutation of the order of the 15 pieces can only be obtained if the empty square is not moved or moved two rows, and an odd permutation of the order of the 15 pieces can only be obtained if the empty square is moved one or three rows.

This small tweaking with the conventional stuff is a brilliant example of how small things can have complex consequences.

Vansh