The Game of Fifteen

The Game of Fifteen

June 11, 2020 3 minutes reading time Development javascript

How to play

The Game of Fifteen is a sliding puzzle that consists of a square with tiles numbered from 1 to 15 in random order with one tile missing. To solve the puzzle, you must place the tiles in order by moving tiles to the empty space.

Play the game online by clicking/tapping on the tile you want to move. Only tiles neighboring the empty space can be moved. Once you solve the puzzle, the tiles will not move anymore. Just play a new game by clicking the button. I can guarantee you that every game presented here is solvable. Read below for an explanation.

Hint: if your browser doesn’t display the board correctly, try clearing your browser cache and try again.

Parity of a permutation

For the Game of Fifteen, a permutation of the numbers 1 to 15 is a state of the game if read line by line - like a book. Some smart people have figured out, that an even parity (is dividable by 2, like 42) is an invariant of a solvable permutation of the Game of Fifteen. In computer science, an invariant is a condition that is always true, i.e. doesn’t change for a section of code.

The parity of a permutation is simply the number of inversions. An inversion happens when an element with a lower index has a higher value than a value at a higher index.


  • 1, 2, 3 has no inversions, all elements are in order
  • 2, 3, 1 has two inversions, because 2 is greater than 1 and 3 is greater than 1
  • 1, 3, 2 has only one inversion, 3 is greater than 2

There are about 1.3 Trillion (Billion for us people in Central Europe) possible permutations of the numbers 1 to 15. Only half of them have an even parity. When we calculate a random permutation for the start of a game, it then makes sense to filter out all the permutations with an odd parity. Because we do not want to present an unsolvable game.

Here is a fragment of source code that calculates the parity and checks if it’s even for the examples from above.

// for (let i = 0; i < limit; i += 1) {...} vs
// range0(limit).forEach((i) => {...})
const range0 = (limit) => [...Array(limit).keys()];

const isEvenPermutation = (p) => parity(p) % 2 === 0;

const parity = (p) => range0(p.length)
  .map((i) => range0(p.length))
    .filter((j) => i < j && p[i] > p[j])
  .reduce((agg, v) => agg + v, 0);

// true, 0 inversions
console.log(isEvenPermutation([1, 2, 3]));
// true, 2 inversions
console.log(isEvenPermutation([2, 3, 1]));
// false, 1 inversion
console.log(isEvenPermutation([1, 3, 2]));

Source code

I have created this GitHub Gist with the complete HTML page including the JavaScript source code. It runs in the browser and uses just Tailwind CSS for styling and plain ES6-style JavaScript. The source code can be used for all square board sizes, not only 4 by 4. You have to adjust the boards GUI or dynamically generate it - it’s easy.