Image-Guided Maze Construction Research Paper

September 23, 2023

I was doing some exploring as I wrote a series of articles on the use of AI and creating mazes and found a paper from July 2007 on Image-Guided Maze Construction. The authors were Jie Xu and Craig S Kaplan from the University of Waterloo. Basically they looked into creating maze art using algorithms and computer graphics. They did a wonderful 25 minute video explaining the concept which is worth watching. You can find the paper and the original video at this link.

What does the paper say ?

The paper "Image-Guided Maze Construction" by Jie Xu and Craig S. Kaplan presents a technique for automatically synthesizing pictorial mazes from images. The system allows the designer to manually partition an image into a set of regions and assign style parameters to each region. They are also able to sketch a schematic layout for the maze's solution path. The final maze will contain intertwined passages in each region, connected together by breaking walls between regions.

The paper describes four different maze textures: directional mazes, spiral mazes, random mazes, and user-defined lines. Each texture is controlled by specialized parameters. The paper also describes an algorithm for building mazes that respect the layout of a given solution tree.

The system has been implemented as an interactive application. The paper presents several examples of mazes generated by the system, including mazes based on photographs of the Great Wall of China, a portrait, and a discus thrower.

So why do I bring this to your attention so many years later ?

Well because of these 3 blog posts I wrote:

And when you combine the ideas of those 3 posts with this research paper and I come to the following conclusion: AI will be reading things like this paper and will be making great mazes soon. I can image someone like these authors working with AI and teaching it to make maze art relatively quickly IF they want to take the time to do it. Will it change what this site looks like ? Only time will tell. Maybe it becomes the typewriter in the year 2000. Let’s hope not !

Side note: I used Starryai to generate the picture for this blog using this prompt:

Make me a logo for a blog post titled Image-Guided Maze Construction paper. This is what it came up with.

Maze Construction - 5 Maze Design Decisions Make All the Difference

May 24, 2023

I’ve taken the time to look at 40 different maze constructions, giving you step by step Instructions on how to make each type. I had a fun time with the project but wanted to expand on the subject with a more detailed look at what the key differences are between each type. Also, in my old job I spent like 90% of my time making spreadsheets and this is a perfect time to break one out ! No V or XLOOKUPS though.

First let’s define a few common maze terms:

Start - Where you begin a maze from.
Goal - Where you need to arrive at to properly solve the maze.
Wall - The portion of the maze that creates uncross-able boundaries
Pathway / Paths - What you travel along in the maze to solve it, typically the place between walls
Outline - The outer wall of the maze or border. Can be a simple shape or something more complex like a picture.

So what are the key items that set the Maze constructions apart from one another ? I believe there are 5.

5 Maze Design Decisions that Differentiate Maze Constructions:

Where the maze starts
What type of pathways/walls the maze uses
Does the maze have an outline
Are there any conditions you need to follow to solve it
What is the purpose/goal of the maze

While there are other things that differentiate between mazes, these are the core 5 items. Let’s take a look at each one with some examples.

Where the Maze Starts
I believe there are 5 categories of how you might start a maze:

Border Start
Internal Start
Undefined location
Multiple locations
Specialty options

I would say that most mazes start at the border of the maze on the outside of the main maze structure. Essentially at the border you find an Arrow directing you into the structure or the word Start. But, there are more options. The internal start is the next most common. Escape mazes, by definition start inside the maze, but Arrow Mazes and Slide Arrow Mazes also start internally within the maze border. Block Arrow Mazes. Number Mazes, and Letter Mazes can also start internally depending on how you structure your maze. The third main start is the undefined start, usually with an undefined goal, meaning the start is either not known to you or it is selectable. A good example is the Dual Entrance/Exit Maze where you select the direction you wish to solve the maze in. While a Start/Goal type Arrow Maze gives you 2 options and you need to discover which is the start and which is the goal ! Finally we have 2 special cases, the Line Leads Maze where the start is a variety of multiple options, and the Pipe Maze construction where the start is the faucet handle (not really internal or border, but kind of both). Read about 9 common Options for Starting and Ending a Maze.

Here are examples of each type:

**Undefined Start** - You choose the start and the goal

**Undefined Start** - Start / Goal Arrow Maze - You determine which is which !!

**Multiple starting options** - Line Leads Maze

What type of Paths/Walls it Uses

I believe there are 5 categories of how you might construct pathways and walls:

Walled Pathway
Gridded Specialty
Grid Walls
On line Pathway
Specialty pathways

The paths are integral in the construction of any maze. The thing about paths is they are created usually by drawing walls. They really don’t exist without the other. The maze you are picturing in your head is probably the most popular pathway, the walled path, typically a line. Those lines create walls that you travel between. The next most popular pathway is the gridded specialty structure. It encompasses all of the maze types that use box grids to house the maze. Think of Arrow, Slide Arrow, Number, Letter and Puzzle Mazes. Each uses a grid pattern to house specialty symbols/items that dictate how you may move. Finally, the standard grid wall structure. A regular grid wall structure allows movement in a more typical way, through cuts in the walls to create a pathway. The last common pathway is the on line pathway. In this type you travel on the line, so that there are no walls. This exists for On-Line, Line Leads, Conditional Path, and Directional Mazes. The last group are a variety of specialty paths each unique to 1 or 2 maze types. 2 specialty paths share the common design of weaving, the Pipe maze and the Weaving maze., each which allow the solver to move over and under the other pathways! Additional specialty mazes include Pattern Mazes which will vary in their pathways based on the pattern you are mazing. Object Mazes use objects for walls, creating unique pathways. Draw and Cut mazes are similar in that the pathways reflect whatever is is you have drawn. For additional reading, I did a blog post about the 12 most common pathway types: Maze Pathway options.

Let’s look at some examples for each:

**Specialty Pathway** - Objects (words in this case)

Does the Maze Have an Outline

As far as constructions go this is pretty simple because it is a Yes/No answer. Bordered mazes are the standard with defined outer edges common to determine the shape and size of the solvable maze. However, some mazes have less structure. Think about a Weaving Maze and a Pipe Maze. Neither has a defined border as you travel in pathways which create the structure of the maze. Non-bordered mazes are also easy to see with Line Leads Mazes, where the squiggly lines make the structure chaotic. Some Mazes have defined structures, but no real border like a Math maze. You could make a math maze in a defined bordered structure, but it is not pertinent to solving.

Are there any Conditions you need to follow to solve it

Most mazes are rather straightforward in how they are solved, but there can be a LOT of variety in the conditions. So the answer is either Yes or No, but the Yes has a huge amount of different conditions ! Let’s start with a standard maze which has No conditions to solving. The solver enters the maze, travels in it, then exits at the goal. That is the normal flow of most mazes. But sometimes there are conditions. A common clue is that if the maze comes with directions it is most likely conditional. Let’s look at a few: Solve the Puzzle to Move can describe many conditional mazes generally like a Block Move Maze. A Block Move Maze requires you to find a path between the start and goal by finding and moving the appropriate blocks to create a path. There are rules on how those blocks may move. A Puzzle Maze requires you to solve the puzzle to move correctly. A Number Maze requires you to choose the correct given numbers to move correctly. Same for a Letter Maze. A Conditional Path maze is by definition part of this as the puzzle you need to solve determines your movement. Likewise a Directional Maze (like a left turn only maze). Next is the Non Repeating Path condition. This is commonly used in a Find Items Maze and a Go & Return Maze. Each or these maze constructions require you to visit places in the maze before exiting, but the path you take may not backtrack from the way you came. Finally there is the Allowable Movement maze types. For me this includes Arrow Mazes and Slide Arrow mazes where movement thru the maze includes options on WHERE to go, but not HOW you may move. For more examples check out my blog on the 11 most common conditions: Conditional Pathway options

Let’s look at some examples:

**Solve the Puzzle to Move -** Left Turn Only

What is the purpose/goal of the maze

I believe there are 5 categories for the purpose of a maze

Reach the Goal
Escape the Maze
Solve a Puzzle
Specialty purposes
Unsolvable mazes

This answers the question - What are you trying to do in order to solve the maze ? The most basic purpose of a maze is also the most common, to reach the goal. While you may think this is the case for all mazes, here are a few other options: Escape the maze - this is of course used for the Escape Maze, where you start inside the maze and your goal is to escape the outer border. There are also instances where the purpose of the maze is to Solve a Puzzle. This includes a Hidden Trail Message Maze where the act of solving the maze creates a message for the solver. It also includes the Find Items Maze type where reaching the goal is only half of correctly solving the maze, you must also find certain items along the way ! Finally there are some Specialty purposes that only apply to 1 or 2 maze types per instance. Like the Math Maze whose purpose isn’t really to reach the goal or solve the puzzle, but you do each in order to learn mathematics ! And there is the specialty maze that should be used only for April Fool’s Day, the Unsolvable Maze. Or you made a mistake when you designed it, oops.

**Solve a Puzzle** - Hidden Message Trail Maze

**Solve a Puzzle -** Find items, then exit

So these are my 5 Maze Design Decisions that Differentiate Maze Constructions with examples. I have taken the time to summarize each of the 40 maze constructions based on these 5 criteria. Check out the chart below. Yellow highlights are on things that are not typical for a particular category.

Of the 40 mazes construction types, if we select the most common type for each of the 5 most common characteristics how many maze types are included ? So this would mean a maze that:

Starts at the border
Has walled pathways
Has an outline
Is NOT conditional to solve
And whose purpose is simply to reach the goal

That leaves us with 12 closely related mazes out of 40. And, if we wanted to add a 6th differentiator, the differences between these 12 mazes are generally HOW the walls/pathways are drawn. Happy maze designing !

Maze options - Conditional Path options

April 11, 2021

Welcome to Part 3 of my 3 part series on maze construction options. Previously, I gave you step by step instructions to make over 40 different maze constructions. But making a maze also means making a series of different decisions, each that will change the look and difficulty of the maze. Here are the Parts we will cover in the series:

Part 1 - Starting and Ending a Maze

Part 2 - Maze Path options

Part 3 - Conditional Path options

Let’s get started with Part 3: Conditional Path Options

In Part 1 we looked at starting and ending your maze. In Part 2 we looked at all the different options for making your pathways. Today we look at a specific subset of maze construction that I call Conditional Path Mazes. This group of mazes is a choice you make as a maze maker to give your solver a bit of an extra challenge. Call them what you want, but can they follow a set of guidelines, conditions, or rules to solve the maze ? Let’s look at types:

1 - Find Items Maze

This is a simple condition mostly. You move through the maze from start to finish, but along the way there are places you need to visit to essentially find items, thus the name. It also works well for Map mazes, where finding the item, or visiting the city is the goal of the maze. Here’s a Germany maze for example:

Note: A subset of this version would be the find items in order maze, where you stipulate the order things must be found/visited.

2 - Avoid Items Maze

The opposite of finding is avoiding, at least for maze construction. This is a natural option for mazes, after all you are already avoiding dead ends !! Really these Avoid Items just create artificial dead ends within the maze, and hopefully help tell a story. Like this example of a Fish avoiding hooks. I could have made more walls but what fun is that ?

3 - Directional Mazes

A maze that states the directions you may move in is a directional maze. To reference Part 2 in the series on pathways, this could be a regular pathway or online. Below is an On Line Maze with the condition that you can only turn left to solve it.

4 - Puzzle Maze

The condition of this maze type determines your movement. This means solving a puzzle to determine your movement thru the maze. In the first example below the movement is determined with a key - you must follow the directions to solve the maze and there are no choices, but with the 2nd puzzle movement is determined by rules - and there are multiple possible ways to move between grids.

5 - Letter Maze

Simply the rules involve letters…and in this version you have choices on where you move…i.e. this M or that M from START.

6 - Number Mazes

Or similarly, numbers to determine movement. This particular variation is based on division and is a Counting maze.

7 - Color Grid Maze

Movement thru the maze is determined by following a series of colors in order.

8 - Conditional Color Path Maze

Similar to #7 but the pathways change colors rather than being colored blocks. As someone who makes a lot of mazes these are both fun and challenging to create. The non grid format gives you options to get complicated and math-y !! If you are a maze maker I suggest you try to make one of these.

9 - Non Reusable Pathways

This is used in a Go & Return Maze and sometimes in Find Items Mazes. The solver is asked to move thru the maze and never visit the same place more than once while solving. These tend to have “open floor plans” when it comes to walls. You want freedom of movement and choice to complicate the solve.

10 - Go & Return Maze

A Go & Return Maze is part #9 Non-reusable Pathway and part #1 Find Items Maze combined with a Start/Goal that is the same. So maybe this is a combination of things we have already covered, and maybe it isn’t. I thought it was unique enough to get it’s own option. In a Go & Return Maze you go from start to an internal part in the maze, then return back to start without using the same path. Those are the conditions to solve ! Here is an example:

11 - Math Maze

A variation of the number maze where your movement is determined by needing to solve a math problem to correctly advance. If you are interested I have made 11 different Math Maze Templates that are free to download.

You may also like:

What I use to make my mazes

Using Color for Maze Pathways - A Case Study

Maze Design Case Study - Designing a Grid Maze in Different Difficulty levels

How to Make a Math Maze

Maze options - Maze Path options

April 8, 2021

Welcome to Part 2 of my 3 part series on maze construction options. Previously, I gave you step by step instructions to make 40 different maze constructions. But making a maze also means making a series of different decisions, each that will change the look and difficulty of the maze. If you missed Part 1, I suggest you start there !

Part 1 - Options for Starting and Ending a Maze

Part 2 - Maze Path options

Part 3 - Conditional Pathway options

Let’s get started with Part 2: Maze Path Options

In Part 1 you figured out how you will be starting and ending your maze. Today we will explore some options for how you make your pathways. Pathways are of course where the maze solver travels as they are moving thru the maze. We start with the most common:

1 - Walled pathway

The most common pathway you will find. Self explanatory to anyone who has ever done a maze and most people that haven’t. The solver travels between walls made of lines. Here is what it looks like:

2 - On Line Pathway

On line pathways are exactly as described - the solver travels on the lines and not in between them. This defines what an On Line Maze is, but can also used for Directional Mazes, which would have instructions such as - solve this maze by only turning left. Here is an example of an On Line Maze. If a solver is not paying attention, they may try to solve this maze as if it was using walls. If you use this construction I suggest adding directions explaining the “on-line” aspect.

3 - Weaving Paths

Weaving Paths are intertwined and allow the solver to travel over/under other paths within the maze. Some use shadows or arrows to better define the movement and enhance the look of the maze, but generally it looks like below:

4 - Themed Pathways

This category is a catch all for a variety of themed pathways you find and become common in children’s maze and activity books. Some examples of these include Railroad tracks, a Roadway, a Subway Line, Sidewalks, and Pipes. Really places where people would naturally travel, or in the case of Pipes, water would. Here is a pipe maze example (which happens to also be a weaving pathway). Notice that I used shadows to further define the weaving aspect.

5 - Grid “Pathway”

Grids can used for regular mazes, but are most commonly used in the different types of Puzzle Mazes. Puzzle Mazes, Block Move Mazes, Number Mazes, Letter Mazes, all use grids to give them structure. Below is a Standard Grid Maze and a Puzzle Maze to show 2 different variations of how grids can be used:

6 - Arrow Mazes

While Arrow Maze and Slide Arrow Mazes both use a grid for structure, the pathway you are following is actually determined by the arrow you land on and the direction you choose along the way. I consider this more about the arrows than the grids they are in. Here is an Arrow Maze as an example:

7 - Free Drawn pathways

I say free drawn pathways, but really what you are drawing by hand are the walls. You get it. This encompasses all the variations that don’t quite fit into a perfect grid or grided path system. The distance between walls varies. Here is a crushed beer can (drawn digitally) with free drawn paths:

8 - Background Picture

Mazes can have backgrounds, some of which have nothing to do with solving the maze. Think of making a maze over a picture of something. You could use the border as the Maze, but internally the picture only helps to decorate the pathways. Here is my maze from National Book day for example:

9 - Objects as Walls/Paths

Similar to what I mentioned before above, this is really about creating pathways by what you use as walls. Using objects or words for the walls creates a very interesting path for the solver. I think they are fun to make and you can be very creative with it. Here are examples of pathways that were made using circles and many small boxes:

10 - Patterned walls

Drawing a standard maze with straight lines as walls can get boring for both the mazer and the solver. This can be made more interesting by using a pattern that repeats for the walls. Like a simple back and forth, or a curve. Notice the example below how a simple maze transforms when drawn with standard walls versus with patterned walls. Very different looks for the same maze.

11 - Colored Paths (walls)

There is an option to enhance the maze you have made using colored walls to define sections. It is much more common to see color used as a background, but keeping this as an option can really enhance your maze making. The example shows a wine maze, specifically a red wine maze as defined by the colored walls ! You might want to read about a case study on colored pathways.

12 - Path Spacing

While most pathways you come across will be walled, there are different looks achieved ( and different maze types resulting from) based on how the same lines are spaced. In the examples below we have walled mazes used to maze the same object with 3 different looks. #1 is a Kids maze with highly spaced walls. #2 is a standard maze with normally spaced walls. #3 is a standard equal where the walls are pathways are of equal width.

3 mazes of Ohio with different maze path spacing

And while this is not the best example, below is a design that used different path widths for each different section of the object. I should mention that the background color is the same in every section. Notice how the bishop has a very dark colored head because of the closer pathways with a lighter core section. Line spacing is a great way to change the color and shading of an object you are mazing !

13 - Use the Drawing as Walls (Natural shape of the subject)

When making maze art I find that the best looking mazes are ones where I don’t create any pathways, but where the pathways are naturally created by the object I am making a maze of. Typically this involves the draw and cut method of maze making where I draw something, then cut a maze into the structure. Almost every building maze on this site uses that method. In the example below of Museo Soumaya the panels of the building become the walls. Panels, bricks, windows, they all have become pathway makers for me !