( i is red) and ( i is not green) and ( i is not blue), orĮxactly two coloured edges of the same colourĮdges in the graph represent connections between cells in the puzzle.( i is green) and ( i is not blue) and ( i is not red), or.( i is blue) and ( i is not green) and ( i is not red), or.So for edge i, we get an expression like: Of course, only one of these can be true, which brings us neatly to … Constraints Each edge must have exactly one colourĪs mentioned, this constraint follows on naturally from the variable definition. Variablesįor example, in a 3-colour map, edge i may be associated with three separate variables: SAT, such that the solved expression (i.e. all the variables are set to “true” or “false”, and all the constraintsĮvaluate to “true”) will give us a valid puzzle solution 3. In order to convert a given Flow Free level to SAT, we must come up with a set of variables and constraints for Scratch, we can instead convert this problem into a Boolean formula, and pass it off to a pre-existing solver. So instead of having to write our own solution from This is useful because this particular problem is very well studied, and as a result, there are many extremelyĮffective off-the-shelf libraries for solving SAT problems. If this is the case, the formula is called satisfiable. In such a way that the formula evaluates to TRUE. … asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE Nodes and edges, depending on the exact map.Īs I have alluded to already, one convenient way to solve a puzzle like this is reduction to Boolean While with alternate shapes, we would have to employ some combination of adding/removing For walls or blocked connections, we would Horizontal connections and the other with the vertical connections. Or in the case of the “bridge” puzzles, we would have to split any bridge cells into 2 nodes. Puzzle simply require an additional edge: For example, puzzles with additional connections, like this warp However, this also works for the other variants. These are marked by a colouredįor example, for a square map this looks like: We also mark out which cells are “terminal” cells, where flows must start and end. If a pipe can move from one cell to another, then those two cells will be connected by an edge. Each cell is represented by a node in the graph. Representing a puzzle map as a graph is relatively straighforward. One way of representing any given Flow Free puzzle is a graph 2. So what we need is something that can handle these cases as well as the standard square map. Or, of course, some combination of the three: These can be grouped according to a few key characteristics: Alternate shapes The Flow Free app comes with a range of additional maze layouts, some of which would be cumbersome to store in a Handle everything that Flow Free has to offer. This is a fine approach, and will work for the examples we’ve seen so far. The goal of the puzzle is then to find the correct (more on this later) colour for Perhaps the most obvious way to think about the puzzle is as it is displayed in that app. However, in order to pass the puzzle off to a computer to solve, we need to represent the puzzle as data. The goal of the game is to connect the dots with pipes that can move to adjacent cells, such that every cell is Get harder by either increasing the size of the map, and/or increasing the number of coloured dot pairs. As you progress through the game, the levels This usually (although not always) takes the form of a square grid. The core mechanics of the game are pretty simple 1.Īt each level, you are presented with a map with pairs of coloured dots occupying some cells: This blog explores one way to do this using Boolean Satisfiability (SAT). There is a feeling of satisfaction that comes with solving one of the game’s puzzles.īut wouldn’t it be more satisfying if we could find a solution to all Flow Free maps? And to be fair, I can see the appeal (to an extent…). A lot as in everyĭay for the past few years. On a recent holiday, I found out that the friend I was travelling with plays this game a lot. Torvaney | Flow Free: A SATisfying solution Flow Free: A SATisfying solutionĪ solver for the Flow Free puzzle game using Clojure and SAT.
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