All human–created systems––e.g., educational, economic, legal––have a design. These design characteristics vary, with some aspects being welcoming, beneficial; other aspects, not so much. We tend to focus on design flaws because of how easily they intrude into our consciousness and make themselves known.
A maze is a path or collection of paths, typically from an entrance to a goal. The word is used to refer both to branching tour puzzles through which the solver must find a route, and to simpler non-branching (“unicursal”) patterns that lead unambiguously through a convoluted layout to a goal. (The term “labyrinth” is generally synonymous with “maze”, but can also connote specifically a unicursal pattern.) The pathways and walls in a maze are typically fixed, but puzzles in which the walls and paths can change during the game are also categorized as mazes or tour puzzles.[1]
Consider product design: *You purchase an expensive item that has a crucial component made of plastic. Invariably, after many uses, the piece of plastic breaks, rendering the entire device unusable. *A refrigerator has an ice cube dispenser built into its outer door that’s too wide to accommodate a standard size glass. When you press the lever, some of the ice spills out onto the floor. *A mechanical device has many moving parts, some of which snap close to create a pinch point. It’s only a matter of time before hair or finger becomes entrapped, causing injury.
Indoors, mirror mazes are another form of maze, in which many of the apparent pathways are imaginary routes seen through multiple reflections in mirrors. Another type of maze consists of a set of rooms linked by doors (so a passageway is just another room in this definition). Players enter at one spot, and exit at another, or the idea may be to reach a certain spot in the maze. Mazes can also be printed or drawn on paper to be followed by a pencil or fingertip. Mazes can be built with snow.[2]
Poor design, impractical design, dangerous design––we encounter these daily. A lawsuit has many moving parts, any one of which can derail, malfunction, break. So, the question becomes: how to thrive when the design characteristics of a particular process are not optimal. We see this everywhere. Evidence and proof do not necessarily come prepackaged in a form most suitable for our needs. We’re often at the mercy of chance and happenstance.
The mathematician Leonhard Euler was one of the first to analyze plane mazes mathematically, and in doing so made the first significant contributions to the branch of mathematics known as topology. Mazes containing no loops are known as “standard”, or “perfect” mazes, and are equivalent to a tree in graph theory. Thus many maze solving algorithms are closely related to graph theory. Intuitively, if one pulled and stretched out the paths in the maze in the proper way, the result could be made to resemble a tree.[3]
Design shortfalls are efficient and expedient, but also expensive––things do not work optimally because others have made a decision without regard to all probable consequences. There’s a reason for this––once a system works tolerably well, there’s not a lot of incentive to destabilize or disrupt. Other than “planning,” “anticipating the worst,” or “having a backup plan” what else can be done to thrive in non–optimal circumstances?
The wall follower, the best-known rule for traversing mazes, is also known as either the left-hand rule or the right-hand rule. If the maze is simply connected, that is, all its walls are connected together or to the maze’s outer boundary, then by keeping one hand in contact with one wall of the maze the solver is guaranteed not to get lost and will reach a different exit if there is one; otherwise, the algorithm will return to the entrance having traversed every corridor next to that connected section of walls at least once.[4]
Let’s combine two pairs of concepts–-“natural” vs. “artificial” and “effortlessness” vs. “strenuousness.” These terms can be combined in four ways: natural effortlessness, artificial effortlessness, natural strenuousness, and artificial strenuousness.
Another perspective into why wall following works is topological. If the walls are connected, then they may be deformed into a loop or circle. Then wall following reduces to walking around a circle from start to finish. To further this idea, notice that by grouping together connected components of the maze walls, the boundaries between these are precisely the solutions, even if there is more than one solution…[5]
Natural effortlessness. A natural process and ability combine to promote optimal easily obtainable results. Artificial effortlessness. An artificial process and ability combine to promote optimal easily obtainable results. Natural strenuousness. A natural process and ability work at odds to each other. Artificial strenuousness. An artificial process and ability work at odds to each other.
If the maze is not simply-connected (i.e. if the start or endpoints are in the center of the structure surrounded by passage loops, or the pathways cross over and under each other and such parts of the solution path are surrounded by passage loops), this method will not reach the goal.[6]
The first two categories are intuitive, straightforward; the third, a bit more challenging. What’s most troublesome, what gives rise to the most headaches, is the final category: artificial strenuousness. Some person, or group, has intentionally set up artificial maze–like barriers to make our lives more complex. They have done this because they were not interested in our ease, level of proficiency, or degree of comfort––decisions made by others outside our control for a variety of reasons.
Another concern is that care should be taken to begin wall-following at the entrance to the maze. If the maze is not simply-connected and one begins wall-following at an arbitrary point inside the maze, one could find themselves trapped along a separate wall that loops around on itself and containing no entrances or exits. Should it be the case that wall-following begins late, attempt to mark the position in which wall-following began. Because wall-following will always lead you back to where you started, if you come across your starting point a second time, you can conclude the maze is not simply-connected, and you should switch to an alternative wall not yet followed.[7]
Which brings us back to the subject of lawsuits. There’s no question a contested lawsuit implicates at times the phenomenon of artificial strenuousness. How to best adjust? One approach is to take the weaknesses within any given system and somehow convert them into strengths. Rather than anger or frustration, defiance or resistance, look for ways to utilize most optimally that which the system offers. Another approach can best be termed “systemic adaptation”––where solutions to problems within a system can be created on–the–spot from materials not intended or designed for that purpose. A final method is simply to “keep one’s hand on the wall”––that is, to maintain a known frame of reference notwithstanding the absence of clear markers.
It is possible for a maze to have three or more dimensions. A maze with bridges is three-dimensional, and some natural cave systems are three-dimensional mazes. The computer game Descent uses fully three-dimensional mazes. Any maze can be mapped into a higher dimension without changing its topology.[8]
Navigating the Maze. Poor design, impractical design, dangerous design––we should not be surprised. Rather, we should come to terms, make adjustments, seek solutions––and thrive to the extent we can. Remember, the fourth category is that which others have designed––decisions made long ago in response to that which we can no longer see. From time to time, it’s probably a good idea to stand in their shoes and ask why those decisions were made. Once the design characteristics are recognized, then you can better take steps to adapt––adaptation defined as a standard of practice where both strengths and weaknesses of a system are taken into account and in fact relied upon.
[1]Mazes/Wikipedia /https://en.wikipedia.org/wiki/Maze
[2]Mazes/Wikipedia /https://en.wikipedia.org/wiki/Maze
[3]Mazes/Wikipedia /https://en.wikipedia.org/wiki/Maze
[4]Maze Solving Algorithm, Wikipedia/
https://en.wikipedia.org/wiki/Maze_solving_algorithm
[5]Maze Solving Algorithm, Wikipedia/
https://en.wikipedia.org/wiki/Maze_solving_algorithm
[6]Maze Solving Algorithm, Wikipedia/
https://en.wikipedia.org/wiki/Maze_solving_algorithm
[7]Maze Solving Algorithm, Wikipedia/
https://en.wikipedia.org/wiki/Maze_solving_algorithm
[8]Mazes/Wikipedia / https://en.wikipedia.org/wiki/Maze
