Maze Equivalence

"Maze equivalence" is the simple observation that most Mazes are the same type of puzzle. The fundamental type, to which most every Maze can be reduced to, is the "arrow Maze". This is basically just a directed graph in mathematical terms. It can be represented by a set of points or vertices, with arrows connecting them. In solving the Maze you move from a start vertex, to an end vertex, following the direction of arrows from vertex to vertex. More specifically the Maze consists of a finite and easily enumerable set of states, where at each state there's a finite and easily enumerable number of choices you can take to change to a different state.

Many types of Mazes with rules have been invented, however they are all basically directed arrow Mazes. Three good sites for various rule Mazes are by Robert Abbott, by Adrian Fisher, and by Andrea Gilbert. The creativity in rule Mazes comes from the conception of the rules, and the artistry of expressing the various links and potentially large number of states within a relatively simple diagram.

Simple cases: Some Mazes map directly to arrow Mazes. Examples are:

State cases: Some rule Mazes involve state, where not only does the solving object have a position, but it can be in one or more states at each position. This type of Maze can be reduced to an arrow Maze that looks similar to the original too, by having multiple levels, one level for each state the solving object can be in. States often involve direction, i.e. the direction you're facing at any one point affects the choices available.

Complex cases: Things get more complicated when there are multiple things in a Maze that can have state, because each combination of states of those things would need to have its own "level" in the arrow Maze.

Non-cases: Maze equivalence isn't a new type of Maze in itself, but it does help in understanding existing types of Mazes, and hence can be good to consider when inventing new types. We've seen how most things can be expressed as an arrow Maze, however what puzzles don't fit this model?

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