Moves: 0
Time: 00:00
Parity: Even
Distance: 0

Move History

No moves yet

Total Inversions: 0
Estimated moves to solve: 0

Instructions & Theory

How to Play

• Click on any tile adjacent to the empty space to slide it

• Arrange the tiles in numerical order from 1-15, with the empty tile in the bottom right

• Try to solve the puzzle in as few moves as possible

Mathematical Background

• The 15 Tiles puzzle represents a permutation of the numbers 1-15

• Each move is a permutation that swaps the empty tile with an adjacent tile

• Not all initial configurations are solvable - this depends on the parity (evenness/oddness) of the permutation

• This demonstrates a key property of permutation groups

Permutation Groups

A permutation is a rearrangement of objects in a set. For n objects, there are n! possible permutations.

The set of all permutations on n elements forms a group called Sn (the symmetric group), with the operation of composition.

Group Properties:

  1. Closure: Composing two permutations gives another permutation
  2. Identity: There exists an identity permutation that leaves elements unchanged
  3. Inverse: Every permutation has an inverse that undoes it
  4. Associativity: Composition of permutations is associative

Parity and Solvability

The parity (evenness or oddness) of a permutation is determined by the number of transpositions (swaps) needed to create it.

For the 15 puzzle:

  • A puzzle instance is solvable if and only if the permutation has even parity
  • Each legal move preserves the parity of the permutation
  • This is why approximately half of all random starting positions are unsolvable
  • This experiment only generates solvable puzzles