The 15-Puzzle Group

Abstract Algebra · Axiom Academy

REAL WORLD The 15-Puzzle Group Discover why some puzzle configurations are impossible to solve—a beautiful application of group theory and the alternating group A₁₅. In 1879, the 15-puzzle became a worldwide craze. Players slide numbered tiles in a 4×4 grid, trying to arrange them in order. Simple premise, but there's a mathematical secret hidden inside... Click tiles adjacent to the empty space to slide them Goal: Arrange the numbers 1-15 in order, with the empty space in the bottom-right corner. Here's a puzzle that looks almost solved—just two tiles are swapped. Try to solve it! The "Impossible" Configuration Can you solve this configuration? Tiles 14 and 15 are swapped. Seems like it should be easy, right? Every puzzle configuration is a permutation of the numbers 1-15. But not all permutations are reachable by sliding tiles! Swapping two elements changes the parity of a permutation This is an odd permutation (one transposition) Key Insight: A permutation is even if it can be written as a product of an even number of transpositions (swaps), and odd if it requires an odd number. The set of all even permutations forms a group called the alternating group . For the 15-puzzle, this is A₁₅. The symmetric group contains all 15! permutations of 15 elements. Contains both even and odd permutations The alternating group contains only even permutations—exactly half of S₁₅. These are the solvable configurations!

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