Many people first discover mazes as children. The mazes appear in some of the puzzle books given to kids to occupy them during long waits, like at a restaurant table. Similar activities, often including mazes, appear on the backs of cereal boxes.
Not all mazes, thankfully, have such low quality as these. But because they appear so frequently, many just assume from a young age that mazes never can provide more of a challenge per size and number of paths than they had seen. This idea, that the primary way mazes increase in difficulty comes through a simple increase in size and number of path choices, betrays its folly to anyone who has solved mazes for long enough. In fact, one of the most interesting challenges in mazes is their difficulty compared to size –– a talented maze maker can make even a small maze more difficult than it appears. This basic task, of making a maze of a given size quite difficult, takes many forms.
Mazes have great variety –– one can split them into two major categories, not just once, but again and again. Possibly the most important of these is the distinction between mazes made for walking around in –– hedge mazes, corn mazes and the like, where the object of the solver is to transport himself from start to finish –– and those written on paper, where the object boils down to drawing a line connecting the start and finish. Though they are alike at the most basic level, these two types have many technical differences, which makes speaking of both at once tricky, except in an overview. Hedge mazes generally have far less complexity in actual setup; the aerial views of these are almost too simple for cereal boxes. They do not need complexity to confound anyone, because without a good sense of direction, anyone can find themselves lost, no matter their technical skill. The two types require different kinds of skill sets. Hedge mazes, besides calling for a good sense of direction for knowing generally where to aim for, require a reliance on great perseverance and a lot of backtracking to solve them. For paper mazes, the solver sees the entire puzzle at once and needs no directional skills, he aims for a final solution, not an instance of solving it. In other words, he would not look through the maze, backtracking with his eyes several times after taking false paths, and then consider the backtracking part of the final solution. With a paper maze he might also have accidentally skipped over a line, either with eyes or a pencil, and not consider the maze solved until after checking it. This column will primarily deal with paper mazes, as will the rest of these distinctions.
Another distinction appears between the most common mazes –– those with boxy passages to follow –– and those with passages going at all different shapes, angles, and curves:
that is, this as opposed to this.
This is mostly an aesthetic difference, but aesthetics affect the difficulty of mazes a good deal. Boxy mazes usually look very slick, direct from the computer screen (unlike my rather inelegant attempt above) and can confuse the solver by making all the different pathways look similar, if done properly. Freestyle mazes, on the other hand, do not follow such a strict formula, and so allow for a great deal of tricks to lead the eye to the wrong places, as well as move more elegantly at times –– it looks much better to have a straight diagonal path than a zig-zag.
As a third major distinction, mazes can come set on a flat plane –– the kind most people have more experience with –– and with bridges over and under other paths, which are essentially three-dimensional, though still exhibited on a flat surface.
This second type can thread a path going from one edge of the maze to the other and overlapping itself, meaning the knowledge of one path’s position relative to the finish does not help much with finding the solution.
These basic differences between types of mazes will help show the amateur maze solver what to expect, when first going out into the wild world of mazes.
This is my only attempt at a boxy maze. In the end I gave up and put some curves in at the bottom. Every article will have a maze (of varying quality) at the end –– be ready!
Image credit: Rees Pingel