15-Puzzle Calculator

The 15-puzzle is a classic sliding puzzle that consists of a 4x4 grid with 15 numbered tiles and one empty space. The goal is to arrange the tiles from 1 to 15 in numerical order by sliding them into the empty space.

What is the 15-puzzle?

The 15-puzzle was invented in 1878 by Noyes Palmer Chapman, a postmaster in Canterbury, Connecticut. It became popular in the early 20th century and remains a favorite puzzle today.

The puzzle consists of:

A 4x4 grid (16 squares total)
15 numbered tiles (1 through 15)
One empty space where tiles can slide into

The puzzle is considered solved when all tiles are in numerical order from left to right, top to bottom, with the empty space in the bottom right corner.

How to solve the 15-puzzle

Solving the 15-puzzle requires strategic thinking and patience. Here are the basic steps:

Identify the empty space
Determine which tile can slide into the empty space
Move the tile into the empty space
Repeat the process until all tiles are in order

For more complex puzzles, you may need to:

Create intermediate steps
Work on smaller sections first
Use a systematic approach

Tip: Always keep track of your moves. The minimum number of moves required to solve a random puzzle is 80 on average, but some configurations may require more.

Puzzle rules and strategies

Basic Rules

Only horizontal and vertical moves are allowed
No diagonal moves
Only one tile can be moved at a time
The empty space must be adjacent to the tile you want to move

Solving Strategies

Here are some effective strategies for solving the 15-puzzle:

Start by placing the largest numbers in their correct positions
Work from the bottom right corner outward
Create a "frame" around the empty space
Use the empty space to create pathways for tiles

Strategy	When to Use	Effectiveness
Corner Strategy	When tiles are near the corners	High
Edge Strategy	When tiles are near the edges	Medium
Random Move Strategy	When stuck with no obvious moves	Low

Puzzle variations

The classic 15-puzzle has several variations that offer different challenges:

8-puzzle (3x3 grid)
24-puzzle (5x5 grid)
35-puzzle (6x6 grid)
Circular 15-puzzle
3D 15-puzzle

Each variation presents unique solving challenges and requires different strategies.

FAQ

How many possible positions are there in the 15-puzzle?

There are 16! (20,922,789,888,000) possible positions in the 15-puzzle, though only about 10.5 million are solvable.

Is the 15-puzzle always solvable?

No, about 50% of random starting positions are unsolvable. The puzzle is solvable only if the number of inversions (pairs of tiles in reverse order) plus the row number of the empty space is even.

What is the minimum number of moves to solve the 15-puzzle?

The minimum number of moves required to solve a random puzzle is 80 on average, but some configurations may require more.