Why you cannot solve the 15-puzzle by swapping 14 and 15: A simple proof

The author of the Habr article provides an accessible explanation of why the classic 15-puzzle becomes unsolvable when the last two tiles (14 and 15) are swapped. Unlike academic proofs that rely on complex group theory, parity of transpositions, and permutation theory, the author offers an intuitive approach that explains the game's mechanics in simple terms. The text examines the logic of tile movement on a 4x4 grid and demonstrates that, given certain initial configurations, reaching the target order is physically impossible. The article is aimed at those who want to understand the mathematical nature of this popular puzzle without diving deep into abstract algebra. The author emphasizes that the probability of success with a random arrangement of tiles is 50%, and the key factor in unsolvability is specifically the order violation in the final row.