Neutralizator puzzle

When I saw the Numberphile's video about pebbling a chessboard, I immediately thought that a puzzle could be created using that game mechanics. The basic idea was cloning. It was both appealing and simple to implement programmatically.

In the video you can see that the described game is fairly different from Neutralizator. It consists of only one type of active element, which Zvezdelina Stankova calls pebble or clone, and a different objective, namely, escaping from a prison. There is also an infinite board, whereas Neutralizator is played on an 8x8 classic chessboard. On the other hand, Neutralizator has two kind of pieces and the objetive is to annihilate all of them.

Initially I wanted to create the puzzle game with that same objective, but as you can see in the video, it is impossible to achieve. So to be able to play and win, I added a second cloning element type that interact somehow with the first one. I think the most obvious type of interaction to avoid blocking the puzzle game is annihilation: when two different types of clones meet, they cancel each other.

Then there was only one more thing to specify: choosing a metaphor for the puzzle theme. It is not very creative or original, but electron-positron particles work very well!

Two initial particles on the 8x8 chessboard placed at opposite corners.
Starting position of Neutralizator puzzle.
Intermediate state of the puzzle with positrons cloned.
Game mechanics is the same as Stankova's clones.

It is very easy to play this brain teaser: you can keep cloning and cancelling particles happily and think that you are going to solve the puzzle in no time. But be aware of states that will end up in game over without actually solving it. You always need a negative particle to the top right position of any positive particles, and vice versa. In the next example, there is a positive particle with no negative particles above it or at the right.

A state of the puzzle that is not solvable.
Gameover: if you reach this position you can continue playing, but you cannot solve the game!

Solution

The objective of the puzzle is to cancel all particles. This is not too difficult. As with other puzzles at puzzlopia, the aim is to solve the brain teaser with the minimum possible movements, and that's another story!

It turns out that Jaap Scherphuis has found a general algorithm for solving the puzzle optimally for all square boards of side n > 2. (Thanks to astralfenix and ffao for their contributions, too).

Animation of the first steps of optimal algorithm to solve puzzle Neutralizator
Zig-zag Jaap Scherphuis's algorithm: first steps, then repeat with negative elements.

Here are the minimum number of steps required to solve square versions of the puzzle (you can see that the required minimum for an nxn grid is 6(n-2) for n > 2):

Optimal solution lengths

  • Board 2x2 2
  • Board 3x3 6
  • Board 4x4 12
  • Board 5x5 18
  • Board 6x6 24
  • Board 7x7 30
  • Board 8x8 36

Variations

It is always interesting to look at possible variations of a game, and in this case I think it is possible to create quite a few puzzle variations, some of them hard to solve. For example we can modify the initial conditions:

This variation consists of supercharging the initial state
This variation adds particles to the initial state. It seems that there is more work to be done for solving this puzzle.
Play Neutralizator II

If we break the initial equilibrium of positive-negative particles and leave not enough room for cloning process, we may end up with an unsolvable puzzle:

A puzzle variation with decompensated + and - particles.
I don't know whether this puzzle is solvable!

More variations could include adding new types of cloning pieces, or both positive and negative to clone top-right (?). What if one particle requires two of the particles of opposite type to be annihilated? Grid variation: try with hex and tria versions. As you can see, this is a never ending story!

If you have any variation idea, please share it!

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