Puzzle 8

Solvability Conditions

Abstract

This study aims to mathematically investigate the solvability conditions of the Puzzle 8, based on permutation theory and their parities. The Puzzle 8 consists of a 3×3 board containing 8 numbered tiles (from 1 to 8) and a single empty space, allowing the movement of adjacent tiles. The objective of the game is to arrange the 8 tiles in ascending numerical order. Although the challenge may seem purely recreational, there is a mathematical structure that determines whether an initial configuration can be solved or not. Using a formal mathematical model of the puzzle through permutations, the parity of each configuration is examined, and it is shown that this, combined with the position of the empty space, constitutes an invariant that determines the possibility of reaching the ordered final configuration. As a result, it is proven that there are possible board configurations that are unsolvable, regardless of the moves performed. The study contributes to understanding how abstract mathematical concepts, such as parity and invariance, apply to concrete problems and can be explored as a didactic tool for teaching and learning mathematics.