Unicursal Mazes have no branching paths; There are no intersections. There are also no dead ends. With pre-determined ("Determined") unicursal paths, the end point is already placed onto the field of points. An unicursal maze path can't just go anywhere it chooses; There are restrictions on where it can and can not go based upon prior decisions. These 'restrictions' are, in a way, the 'rules' for which a person needs to follow if they are to create a legitimate unicursal maze path.

Where does a rule begin or end? Figuring out what constitutes a 'rule' is not as easy as it sounds.

For example, suppose that you were told that you had to bake a cake. You are handed a set of instructions as to how to bake that cake. How detailed do those instructions have to be for you, in particular, to bake that cake?

Could the instructions be as vague as "Pre-heat oven to such-and-such degrees, put such-and-such ingredients into a bowl, mix, pour into a baking pan, place pan into the oven, close the door and wait such-and-such minutes"? Or do those instructions have to be extremely specific, such as "Step #47: Ensure that the oven is plugged into the wall by following the cord from the back of the oven to the outlet. Step #48: Grip the outlet plug and test the firmness of the connection"?

There is no correct answer to this dilemma and different people would have different standards as to what a 'rule' would be. This dilemma is no different when determining how to create an Unicursal Maze Path.

For our purposes, there are two sets of 'rules': The rules for establishing ("the Rules of Establishment") the possibility of creating a unicursal path and the rules for creating ("the Rules of Creation") the path itself.

Both sets of rules are crucial; Without both sets, an unicursal path can not be created. For instance, if it is not specified how the field of points are set up (a 'rule of establishment'), the rules for creating the path itself become meaningless. If the rules of creation are vague, then it doesn't matter how specific the rules of establishment are because the paths could be mis-interpreted and drawn incorrectly.

In order to differentiate between these two sets of rules, they are going to be labeled differently. The "Rules of Establishment" are going to called 'Axioms' or 'Truths.' The "Rules of Creation" are going to be called just that - Rules.

It is time to refer back to the cake example: When examining multiple cake recipes, these recipes presume that the baker is already knowledgeable, to some degree, about baking cakes. For instance, they know what an oven is. They know how to turn the oven on. They know enough to take any items out of the oven first before you pre-heat the oven. They know enough to wash and dry the baking pan before using it.

Although this knowledge is useful in baking a cake, this knowledge is not used in directly baking that cake; This knowledge establishes the possibility of baking that cake. That knowledge are the Rules of Establishment; They are axioms needed that allow the Rules of Creation to occur.

Just as there are axioms for baking a cake, so too are there axioms for creating an unicursal path. Although these 'axioms' or 'truths' have been addressed elsewhere on this website before, a brief overview of them again is warranted:

• A unicursal path consists of points and segments.
• A segment is the distance between one point and the next closest point that is orthogonally above, below, to the left or to the right of that point.
• The points must be evenly spaced from both vertically and horizontally at the same distance and this collection of points is called a "field."
• A path begins at the start point and ends at the end point.
• A path connects every point to only two segments with the exception of the start and end points.

Hereafter, when rules are discussed, they will be Rules of Creation, those rules directly responsible in creating an unicursal path.

When you ask a person to count from the number one to the number twenty, most people would count sequentially, or 'in order': 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 & 20. They wouldn't 'skip over' a number, repeat one or include one that shouldn't be within that range.

In the terms of rules for creating an unicursal path, what would be the First Rule? How shall it be determined what the first rule is? And how shall it be determined what the second, third or fourth rule is?

Again, different people would have different standards in creating an orderly process in establishing these rules. There is no correct answer but a process still needs to be put into place to allow for the systematic discovery of these rules so that no rules are left undiscovered.

The process that has been put into place is the following:

• The process shall begin at the smallest field and work its way up to the larger fields.

Rule discovery shall begin at a 2x2 field and then proceed from the number of rows and then by the number of columns. For instance, 2x2, then 2x3, then 3x3, then 2x4, 3x4, 4x4, 2x5 etc. so forth.

• Each point in a field shall be traversed in a systemic order, with one direction covered per attempt until a direction is successful, then the process resets itself again at that new point.

With an analog clock, the hands on that clock turn to the right ("clockwise") in a circular motion, starting at '12' and going past '1', '2' and so forth until it returns to '12' again. Essentially, the hands point upward, to the right, downward and then to the left. The four ordinal directions (North, East, South and West) can be reduced into numbers: 1 = North, 2 = East, 3 = South & 4 = West. Or, at its fullest abbreviation, "1234."

At every point, a path that is dictated by the "1234" system shall first try moving one segment to the North ("1"). If it does not succeed, it will then try moving one segment to the East ("2"). When it does eventually succeed in moving one segment to a new point, the pattern repeats itself back, starting with "1."

• As soon as a new rule is discovered, it is applied to all future attenpts.

Once a new rule has been learned, it is immediately applied to all future efforts. There is no need to go back to previous mazes; They succeeded without needing that rule.

• Rules can be modified but those modifications need to apply to all prior applications of that rule.

Rules are not 'set in stone'; They can be modified based upon further efforts in applying that rule. However, for that rule to survive modification, it needs to be applied to all prior appearances of that rule without 'breaking.' Otherwise, that modification is essentially a new rule.

• Each combination of numbers shall be tried in sequential order, from "1234" through "4321."

There are twenty four possible combinations of a four-digit number using only the digits "1," "2," "3" and "4," from "1234" to "4321." All twenty-four combinations will be tried sequentially in order.

• Once all of the 'ordered' attempts have been completed, all of the 'unordered' attempts will be completed in the order that they were discovered.

It is naive to think that all of the rules for creating unicursal mazes may be uncovered through these twenty-four possible combinations for each given field and original pair. After all, with larger fields (such as a 5x5 field), there are more than twenty-four unique original paths for each original pair. Within these fields that were not created through the ordered process, there may be new rules that need to be discovered. After all, the purpose for finding all of these rules is to have a set of rules that allows for the creation of ANY unicursal maze regardless of the field or the original path.

Therefore, it is important to create Rule Maps for those paths that were created without an orderly process ("1234" through "4321"). These maps are created differently, with the path 'predetermined' through the normal creation process as described in the "CYCLES" learning module. These paths are not necessarily orderly (although created through an orderly process different than creating a Rules Map) and are called "Unordered" to differentiate between the two types of Rules Maps.

Each of these maps will be processed in the ordered that they were discovered through the normal creation process. Unlike an Ordered Rules Map, each point shall be evaluated simultaneously in all four orthogonal directions. If the predetermined path already created is not covered by an existing rule, then a new rule has been created. That new rule is not given a number but a letter, starting with "A" and going sequentially to "Z" before starting at "AA" and then again going sequentially to "AZ" (and extending to "BA" to "BZ," and so forth).

• Just as with ordered rules, unordered rules are applied to all future attenpts, both ordered and unordered.

Just as unordered rule maps use numbered rules that were discovered during the ordered discovery phase, ordered rule maps may be use lettered rules discovered during the unordered discovery phase. This includes all unordered rule maps that have yet to be created.

In conclusion, the process for Rules Creation starts at the smallest field and goes through each original pair before proceeding to the next larger field. The ordered maps are created first (sequentially) before the unordered maps are created in the order for which they were discovered. Rules discovered in the Ordered Map phase have numbers; Rules discovered in the Unordered Map phase have letters.

The above explanation is not the definitive explanation for Rules Generation and further refinements for this explanation are expected. However, this explanation should give the uninitiated some degree of knowledge in how the Rules for creating Unicursal Mazes are discovered and classified along with their reasoning. Thank you.