Sliding Puzzle Solvable?

I’m helping develop a new game concept, which is based on the sliding puzzle game. The idea is to randomise the initial configuration of the puzzle. However, I quickly discovered that half of the resulting configurations were not solvable. Not good! Here are two approaches to getting a solvable puzzle:

  • build it (by randomly moving tiles from a known solvable configuration) or
  • generate random configurations and check whether solvable.

The first option is obviously more robust. It’s also a bit more work. The second option might require a few iterations, but it’s easy to implement.

I’m going to embrace my inner sloth and go with the latter.

Puzzle Representation

Rather than representing the puzzle as a matrix it will make sense to us to “unwrap” the puzzle into a vector. So, for example, the final configuration of the puzzle would be

[1, 2, 3, 4, 5, 6, 7, 8, 0]

where 0 has been used to denote the empty square.

Inversions and Polarity

To make sense of the algorithm for determining whether a puzzle configuration is solvable we need to define two terms:

  • inversion — an inversion is any pair of tiles that are not in the correct order; and
  • polarity — is the total number of inversions even (solvable) or odd (not solvable)?

Consider the following puzzle configuration which has six inversions:

[1, 3, 4, 7, 0, 2, 5, 8, 6]

Let’s look at the inversions (since the 0 is just a place holder it’s not considered when finding inversions):

  • 3 > 2
  • 4 > 2
  • 7 > 2
  • 7 > 5
  • 7 > 6 and
  • 8 > 6.

Since there are six inversions (even polarity) this configuration is solvable.

Here’s a configuration which is not solvable:

[2, 1, 3, 4, 5, 6, 7, 8, 0]

There’s just a single inversion (2 > 1), so the polarity is odd.

Some notes on the link between inversions, polarity and solvability can be found here.

Python Implementation

The game concept is being implemented using Unreal Engine, a paradigm with which I am completely unfamiliar. So it made more sense for me to implement the solvability algorithm in Python and just pass this on to the Unreal developer.

So here’s the simple function:

def solvable(tiles):
    """
    Check whether a 3x3 sliding puzzle is solvable.

    Checks the number of "inversions". If this is odd then the puzzle configuration is not solvable.

    An inversion is when two tiles are in the wrong order.

    For example, the sequence 1, 3, 4, 7, 0, 2, 5, 8, 6 has six inversions:

    3 > 2
    4 > 2
    7 > 2
    7 > 5
    7 > 6
    8 > 6

    The empty tile is ignored.
    """
    count = 0

    for i in range(8):
        for j in range(i+1, 9):
            if tiles[j] and tiles[i] and tiles[i] > tiles[j]:
                count += 1

    return count % 2 == 0

Let’s give it a test run.

>>> solvable([1, 2, 3, 4, 5, 6, 7, 8, 0])
True
>>> solvable([2, 1, 3, 4, 5, 6, 7, 8, 0])
False

Looks good. No more broken puzzles!

Python
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Andrew B. Collier
Entrepreneur / Data Scientist

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