# 15 Puzzle Game: Existence Of The Solution

This game is played on a $4 \times 4$ board. On this board there are $15$ playing tiles numbered from 1 to 15. One cell is left empty (denoted by 0). You need to get the board to the position presented below by repeatedly moving one of the tiles to the free space:

The game "15 Puzzle” was created by Noyes Chapman in 1880.

## Existence Of The Solution

Let's consider this problem: given a position on the board, determine whether a sequence of moves which leads to a solution exists.

Suppose we have some position on the board:

where one of the elements equals zero and indicates an empty cell $a_z = 0$

Let’s consider the permutation:

i.e. the permutation of numbers corresponding to the position on the board without a zero element

Let $N$ be the number of inversions in this permutation (i.e. the number of such elements $a_i$ and $a_j$ that $i < j$, but $a_i > a_j$).

Suppose $K$ is an index of a row where the empty element is located (i.e. using our convention, $K = (z - 1) \div \ 4 + 1$).

Then, **the solution exists iff $N + K$ is even**.

## Implementation

The algorithm above can be illustrated with the following program code:

```
int a[16];
for (int i=0; i<16; ++i)
cin >> a[i];
int inv = 0;
for (int i=0; i<16; ++i)
if (a[i])
for (int j=0; j<i; ++j)
if (a[j] > a[i])
++inv;
for (int i=0; i<16; ++i)
if (a[i] == 0)
inv += 1 + i / 4;
puts ((inv & 1) ? "No Solution" : "Solution Exists");
```

## Proof

In 1879 Johnson proved that if $N + K$ is odd, then the solution doesn’t exist, and in the same year Story proved that all positions when $N + K$ is even have a solution.

However, all these proofs were quite complex.

In 1999 Archer proposed a much simpler proof (you can download his article here).