Making Difficult Mazes

Most of this site deals with the technical aspects of Mazes. However Mazes have a human side to them as well. Making a difficult Maze is more than just making the Maze large in size (although that certainly helps. :-) Here are other things that increase the difficulty of a Maze:

• Size Ratio: For Mazes of the same cell volume, a Maze with more compact dimensions of a more equal ratio can be harder, because there's a wider area to potentially get lost in. For example, a 25x40 Maze (1000 cells total) will tend to be harder than a thin 4x250 Maze (also 1000 cells). For this reason, 3D Mazes are usually much harder than 2D Mazes of the same volume, e.g. a 10x10x10 Maze (1000 cells again) should be harder than the 25x40 Maze.
• Solution Count: The more solutions there are to a Maze, the easier it is. The hardest Mazes usually have a single solution, or all the solutions are close to each other.
• Solution Length: Often, the longer the solution path, the harder the Maze. This isn't necessarily true, as sometimes the hardest and best designed Mazes have a very short solution, but if you don't know the way you'll most likely wander around through the rest of the Maze. You can at least say that a long solution path guarantees the person will spend a certain minimum amount of time in the Maze, even if they're very lucky. The reverse is true, where having the start right next to the finish makes for an easy Maze of course.
• Curves: An irregular Maze is harder to solve than one on a standard grid, especially from inside the passages. It's easy to keep your sense of direction in a regular Maze, where Mazes with odd angles or subtle curves can get you turned around.
• Loops: Having passage loops or detached walls can make a Maze harder, since it's easy for one to know they're on a wrong path when they hit a dead end, however if they go around in circles they can visit the same path many times before they know they should be somewhere else. Note this doesn't contradict the solution count statistic above. Just have the loops not on the solution path, so they make the Maze harder instead of easier to solve.
• No Wall Following: When solving a Maze, be it a life size version or a Maze on paper, people will sometimes just follow a wall. In the hardest Mazes, this won't work, as you'll just find yourself back at the start again if you try that. To make a Maze like this, have the start or finish in the center of the Maze, and have a passage loop surrounding it. Bridges can be used to disable wall following as well, if a bridge is on the solution path and a passage loop surrounds one end of the bridge. Checkpoints can do the same, if a loop surrounds a wall section in which exists a checkpoint.
• Roundabout Passages: Passages that seem to go one way, but end up sending you somewhere else, make a Maze harder. A good example of this is a spiral, where passages spiral into and back out of its center. In a spiral it's very difficult to see where a passage will lead you without actually following it.
• Unintuitive Choices: Psychologically, people tend to make choices at junctions that lead toward the exit. Having passages that seem to lead away from the exit be the right path can make a Maze harder. Similarly, having the solution path require you to do sharp hairpin turns at junctions can also make a Maze harder, because people tend to be reluctant to "undo" what they just did. Finally, people tend to go right when all other things are equal, e.g. people more often go counterclockwise through amusement parks, presumably because most are right handed or drive on the right side of the road. Having passages that require going left or go clockwise around the Maze can make it harder.
• Repetition: Recognizable landmarks can make a Maze easier to solve, as one can use them as a reference point. Conversely, having different parts of the Maze similar but slightly different can make a Maze harder, as your memory can confuse you more instead of help you.
• Enticements: Landmarks such as bridges, rooms, notes on the wall, and so on, tend to attract a person solving a Maze, as they want to check out the point of interest. Having the solution path go within sight of but not past a landmark is a good way to make many people take the wrong path. A perfect example is one of the plans for Glacier Maze featured a really long passage which turned a corner, where one could see a "clue" sign at the corner. You of course wanted to go down the long passage to read the clue, where once you're there you noticed the long passage was a dead end right after it turned the corner, where the "clue" sign helpfully said "This is a dead end". ;-)
• Deviation Distance: Mazes with blind alleys that deviate a long distance from the solution path make a Maze harder and more complex, because you can go longer before knowing you're on the wrong path. Long false paths also make it harder for one to see which way to go when looking at a map or down on the Maze from above too. Conversely, a Maze with only small hair like dead ends like a pipe cleaner is easy to navigate, where a unicursal Labyrinth is the easiest of all because you're never off the solution path.
• Gender Differences: Although this may not apply to particular individuals of course, in general men and women approach Mazes differently. Men tend to think in terms of absolute location, while women look for and remember landmarks. Hence having or not having landmarks can make the Maze easier or harder for certain genders.

Making Quality Mazes

Just because a Maze is difficult doesn't necessarily mean it's a good Maze. Sometimes the most fun Mazes are relatively easy, where in fact a Maze that's too challenging may just cause frustration. Here are some elements that can increase the fun of a life size Maze. Some of these can also be applied to Mazes on paper:

• Emergency Exits: Safety first of course. If someone panics in a Maze, it's good if they can get out quickly. Fence Mazes tend to have walls that end a couple feet above the ground, allowing one to crawl out if necessary. A nice thing about corn Mazes is you can always get out if necessary by just cutting through the corn.
• Shapes: Making a Maze a picture of something or giving it a theme, makes it a more obvious work of art. It can also create additional public interest, as opposed to the Maze just being an abstract puzzle. A Maze can spell out some advertisement in exchange for sponsorship. A picture or theme can attract the media, where a newspaper or TV station is more likely to do a story and take an aerial photo of the Maze if it actually forms an artistic picture. A picture can be appreciated when you're inside the Maze too, as opposed to only from the air. While going through it, it's a good exercise to mentally connect the passages where you are to the picture, e.g. this triangular shaped room is the cow's ear.
• Ground Cover: It's nice if an outdoor Maze has the ground covered with bark or gravel. With ordinary dirt floors the Maze will be a mess after it rains!
• Map: Many life size Mazes give out maps of the Maze to visitors, or at least have aerial pictures displayed by the entrance or at locations inside the Maze. People who like a challenge are free to not look at the map, while people who need the help or need to get out quickly can use the map. Finding an efficient route on a map and then following it is a good mental exercise.
• Landmarks: Having points of interest in the Maze can increase the fun or at least break up the repetitiveness of all the passages. Landmarks can be checkpoints, bridges, open spaces, signs, and more. Glacier Maze has, in addition to checkpoints and bridges, funny Far Side cartoons on various walls, along with "clue" signs at various points (which may or may not be helpful).
• Bridges: Bridges are nice since they add a 3D element to the Maze, and are a great place to take overview pictures of the Maze from. Of course, they're more expensive to construct, since they need to be sturdy, where they may have a dozen people hanging out on them. I've seen "poor man's bridges" in life size Mazes before, which are basically crossroads where a sign says the rule is you have to go straight.
• Checkpoints: Having checkpoints in the Maze, i.e. things you need to find in the Maze before the exit, can add interest. Checkpoints break up the monotony, where you can feel like you're making progress in stages. With a single solution, it's all or nothing, where there's nothing else to do along the way. With checkpoints, if you're bad at Mazes or in a hurry you can at least say you found the first few checkpoints. When there are multiple checkpoints, the user tends to find the first ones faster, since there are more available when you start. That can engage the user's interest and make them want to find the rest. Checkpoints don't make a Maze any easier, as once you've found all but the last checkpoint or two, the Maze becomes as hard as a normal Maze with only one goal. A Maze of a given size with checkpoints is harder, since you need to cross the Maze several times in order to find them all. Sometimes checkpoints can also be landmarks, e.g. Glacier Maze has four checkpoints in towers at the four corners, where those towers can be seen looming over the walls.
• Ordered Checkpoints: Checkpoints in a Maze can also be ordered, meaning you have to visit them in a particular order. This basically makes the Maze a sequence of Mazes within the same passages, where this challenges a smart person to use their memory to more quickly navigate the paths. For example assume you've found checkpoint #1, however along the way you passed checkpoints #2 and #4. Can you remember the way back to them?
• Separate Mazes: Instead of having just one big Maze in the available space, you can have a few Mazes. For example have a hard Maze for people that like a challenge, and an easier Maze for children or people with less time.
• Loops: Loops in a Maze can be good since they help avoid traffic jams. People won't get bunched up in dead ends, since you can always move forward, or leave any location by two paths. Carpinito Brothers corn Mazes tend to have their checkpoints on cul-de-sacs, since checkpoints often have a group of people hovering around them. I once did a life size tarp Maze which was "braid", i.e. had no dead ends at all.
• Changing Solution: Finally, if your Maze is permanent (as opposed to existing for just a season like a corn Maze) you can periodically change the way through. This can attract repeat business from locals or people who have done it before. Glacier Maze changes the plan of their fence Maze about once a month, where they post the shortest time someone has managed to solve each plan, as a record to try to beat.

Life Size Solving Techniques

So you've entered life size Maze! Here are a few tricks on how to effectively solve one:

• Restroom: First and most important, always visit the restroom before entering a life size Maze! You may be in it a lot longer than you expect. ;-)
• Equipment: Bringing items inside the Maze can help you solve it. A compass can help if you tend to get turned around, and can't use the sun or tall landmarks to orient yourself. Corn Mazes, especially in wet states like Washington, can get very muddy, hence wearing boots, especially rubber boots, is often a good idea.
• Teamwork: If you're solving a Maze with someone, you can work together. For example one person tries one way and the other another, where they can call to each other whether they see a dead end or a continuing passage. Just don't get too far separated from each other! Even if you're doing the Maze on your own, you can talk to or just watch other parties in the Maze. If you see someone go around a corner, then a few seconds later come back, that passage is probably a dead end.
• Look Down: The correct path through a Maze tends to be more worn. In Mazes with dirt floors such as corn Mazes, at a junction the dirt leading down the correct path will often be more packed and with less vegetation. In Mazes with gravel floors, the gravel will often have a deeper path through it.
• Go to the Light: You can sometimes tell a passage will be a dead end before you see the block at the end, because the passage will get darker, due to visible shadows or the block cell being surrounded by walls on three sides. This most often can work in fence Mazes, which have solid and narrow walls.
• Speed: Getting through a Maze quickly often involves just moving as fast as possible (speed walking if the Maze doesn't allow running). When I broke the record for fastest time through one of the setups at Glacier Maze, much of that was accomplished by just going really fast through it, as opposed to being smart. ;-) Solving a Maze quickly is one thing, while solving correctly making a minimum of errors is a different skill.
• Full Minute: If you're timed in a life size Maze, by having a card stamped at the start and end with the current time, and that time is only to the nearest minute, then initially you should get your card stamped right after the machine switches to the next minute. That gives you the full first minute to work on the Maze, as opposed to if your card is stamped right before the next minute, in which case you've already lost a minute a few seconds after you start.
• Changed Solution: In corn Mazes, beware of people carving their own paths through the corn (which can be seen if a passage is narrower or has fallen corn at its bottom) which can make following a map more challenging, since it doesn't fully correspond to the terrain anymore. Hence it's often easier to solve a corn Maze earlier in the year, before it gets too damaged.
• Go around: This should be considered cheating, but it can be used as a technicality in getting to the exit of a life size or paper Maze quickly. If the entrance and exit points are on the outer edge of the Maze, and the objective is merely to get to the end (as opposed to actually going through the Maze) then you can reach the end quickly by simply going around the outside of the Maze! ;-)
• Couple's Test: A life size Maze is a good test for any potential couple! If you and your date can go through the Maze together, and not get angry at each other when you get lost, while you share the decision making, count on happy times together. :-) On the other hand, if your date acts like an angry car driver with "road rage" or insists on making all the decisions where you go, expect them to behave the same way in a relationship.

Solution Notation

• Letters: In my notation, simply use "L" to indicate one should take the first path on the left at a junction, and "R" to indicate the first path on the right. Use "A" to indicate one should go across i.e. take the middle path when at a crossroads. For example, for the traditional plan of the Hampton Court hedge Maze, the way through is simply: "LRRLLLL". For more complicated Mazes that can have five or more passages meeting at a point, use the letters that come after "L" and "R" to indicate the next passage on the left or right. In other words use "M" to mean one should take the second passage from the left, "S" to indicate the second passage from the right, "T" to indicate the third passage from the right, and so on. In all cases if there are an odd number of choices and you want to indicate taking middle one, use "A". Additional letters can be used to describe 3D Mazes, e.g. "U" means to go up, and "D" means to go down. Note "L" really means take the first available passage from the left, even if it turns you to the right, e.g. if a junction has you take either a slight turn to the right or a sharp turn to the right, the slight turn is still indicated by "L" since it's the leftmost.
• Repetition: Numbers can be used to indicate taking a particular direction more than once, e.g. "L4" means take the first left four times in a row. The way through the Hampton Court Maze can be better expressed as: "LR2L4". Parentheses can also be used to mark off a subset of instructions, to be repeated a number of times. For example, with my secret pattern Maze that you can get through by going "left right right", "left right right" over and over, its solution is "(LRR)71R" or "(LR2)71R" which means go "left, right, right" (do that 71 times) followed by one more right to reach the center.
• Reverse: This notation is readily reversible, to navigate from the end back to start. Simply follow the instructions from right to left instead of left to right, and invert the "leftness" and "rightness" of all the letters. In other words, exchange L's and R's, M's and S's, and leave A's alone. The way back out of the Hampton Court Maze is "R4L2R", while the reverse of the secret pattern Maze above is "L(L2R)71".
• Absolute: Note indicators like "L" and "R" are relative to the direction you're currently going, which is the easiest to apply when you're inside a Maze. Additional letters can indicate absolute motion, e.g. "N" means to go North, "W" means to go west, and so on, which can be easier to apply when looking down on a Maze from above.

Maze Formulas

• d = Dead Ends: This is the number of dead ends in the Maze in question.
• j = Junctions: This is the number of simple junctions, where exactly three passages meet in a cell.
• c = Crossroads: This is the number of crossroads, where exactly four passages meet in a cell. For non-rectangular Mazes where more than four passages can meet, have an array of numbers for all types of general junctions, based on how many passages can meet in a cell.
• e = Entrances / Exits: This is the number of entrances and exits, i.e. openings in a boundary wall.
• l = Loops: This is the number of passage loops or detached walls within the Maze.
• i = Isolations: This is the number of isolated inaccessible areas, i.e. collections of passages unreachable from entrances.
• x & y = Horizontal & Vertical Passages: This is the number of passages across and down in a standard rectangular Maze.

• b = Branches: This is the total number of choices throughout the Maze. For rectangular Mazes: b = j + 2c. In general, for each cell increment b by (the number of passages coming together - 2).
• t = Terminations: This is the number of places where you can only go in one direction, i.e. a dead end or an entrance or exit. By definition: t = d + e. For perfect Mazes: t = b + 2. Why is that? Picture a perfect Maze growing like a tree, where you start with a single unbranching passage segment, where t = 2 there for its two ends. Each time you add a branch, that results in a new passage attached to the tree that has a new termination on its far end.
• v = Valence: This is a measure of the "density" of a Maze, i.e. how many wall segments are within it based on what a perfect Maze would have. By definition: v = i - l. Each isolation adds one to this value, while each loop subtracts one. For perfect Mazes, v = 0 of course, although a Maze where v = 0 isn't necessarily perfect (it just means the Maze has an equal number of isolations and loops). For all Mazes: v = (t - b - 2) / 2. Why is that? Solving for t results in a generalized version of the equation above: t = b + 2 + 2v. Each loop subtracts two from the termination count, since it in effect connects two dead ends with each other, while each isolation adds two to the termination count, since it in effect spawns a separate Maze with its initial passage with two terminations.
• n = Nodes: This is the total number of points of interest within a Maze. Nodes are either junctions, dead ends, or entrances. By definition: n = j + c + t.
• p = Passages: This is the total number of passages between nodes. For all Mazes: p = (3j + 4c + t) / 2. Why is that? Each junction is one end of three passages, each crossroads is one end of four passages, and each termination is the end of one passage. That accounts for both ends of each passage, so just divide by two. Another formula for passage count can be found by plugging in the value for t above and simplifying: p = 2j + 3c + v + 1. The way to think of this one, is you start with one passage, where each junction appends two passages to the tree, and each crossroads appends three passages. Loops connect two passages together, decreasing the count by one, and isolations spawn a new passage, increasing by one.
• w = Wall Segments: This is the number of individual wall segments. Any two cells adjacent to each other (the area outside the Maze can be considered a giant cell for this purpose) have a potential wall segment between them. For rectangular Mazes: w = (x+1)*(y+1) - e + v. Note this means if you count the number of wall segments in a Maze, i.e. know w, then you can determine its valence without the complicated process of counting loops and isolations, by solving for v: v = w - ((x+1)*(y+1)) + e.
• o = On Pixels: This is the number of set pixels, in a bitmap picture of a Maze. Assuming walls are one pixel thick of "on" pixels, and passages are one pixel thick of "off" pixels, then for rectangular Mazes: o = (x+1)*(y+1)*2 - e + v. This is basically the same equation above, except that it takes two on pixels to form a wall segment. As above, if you count the number of on pixels in rectangular Maze, i.e. know o, then you can determine its valence: v = o - ((x+1)*(y+1)*2) + e.

Back to Think Labyrinth!