In the case of the puzzle you have identified, you must add a transition between the front and rear regions once the flow is complete because they are not adjacent. If you did not have the transition between the front and rear regions, then you would have had a stranded region. Solving for this stranded region would have been tricky at best.
2. A solver must be able to handle multiple regions. By far, the most common approach to solve this type of puzzle is to first find the single-region solution using a partial flow algorithm. Once the single-region solution has been found, the transitions between regions are then added, one at a time. For example, a solver might add the left and right transitions first, then the top and bottom transitions, and finally the front and rear transitions. One issue with this approach is that the solver must assume that once a transition is added, it does not affect the other transitions. This is not correct. The solver should add transitions in the order that they are added. For example, if a transition affects another transition, then it must be added last.
The failure of the solver to find the solution relies on theassumption that each of the five flows can advance toward itsgoal position at any one time, regardless of the configuration of theother flows; and that each flow can do so without encounteringconflicts with other flows that might be seeking to advance toward thesame goal position. This unduly narrow perspective makes the puzzle solvable.Using the theory of elementary cellular automata, we believe we have identified two “fractal” structures that are responsible for the failure of thesolver to find the solution.
For a cellular automaton to be a “stochastic fractal” (SF), it has tohave a nested set of “stochastic bottlenecks” that are a subset of thesolutions. When the solver encounters a stochastic bottleneck, it has tobegin to calculate the motion of a flow that is approaching thegoal position. But the solver is unable to calculate how that motionwill be affected if another flow is calculated at the same time, and soit abandons the calculation.
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One of the drawbacks to this approach was that we needed to recompile every time we made even a small change to the code. This slowed everything else we were doing down and because we couldn't always have someone waiting for the application to finish building. 827ec27edc