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### Solutions for Act V Scene 4 - Mazes

We are faced with 26 smaller puzzles, or mazes, of dimensions up to 5x5, and one final 10x10 maze, all containing a number of symbols of which their function is initially unknown. Without much else to go by, the logical first step would be to start solving each maze, but unfortunately we have no idea to tell whether our solutions are correct. As we go through the mazes we should try to figure out what each type of symbol means, and use them to build a set of rules that can validate our solutions and are consistent across all mazes. This process can be broken down into four sections:

• Mazes 1-4: An introduction to the core mechanics of the mazes. Solvers experienced with the puzzle concept should find these mechanics familiar.
• Maze 1 is straightforward, introducing the large circle that pertains to the start, and the curved end that suggests the destination.
• Maze 2 resembles more of a maze, but is still quite easy to solve. There exist multiple solutions to this mazes however; the only major way to distinguish the quality of each solution is their length, so the shortest length solution is desirable.
• Mazes 3 and 4 have two starting and ending points. Without imposing some restrictions there appear to be many ways of solving these mazes. It seems that these mazes can be solved from both starting positions simultaneously; moreover, we can enforce a unique symmetrical path where neither of the two paths reach any dead ends or cross over each other. These constraints become more obvious in later mazes where more restrictions come into play (the latter constraint is alluded to in mazes 7 and 8, for example).
The solutions to mazes 1-4 can be found here.
• Mazes 5-8: Adds the mechanic of rings and crosses.
• Mazes 5 and 6 solely feature rings. Two logical conclusions are possible - either we go through every ring or we go through none of them. Both solutions are possible at the present time.
• Mazes 7 and 8 also add crosses. Maze 8 in particular forces any solution to go through both circles and crosses, so only the first conclusion reached above is possible. However, a solution path that traces through every ring must cross at one point in order to reach the destination, so the cross must denote such a point. The previous mazes can be solved with certainty after this realisation.
The solutions to mazes 5-8 can be found here.
• Mazes 9-18: Adds a set of two orange polygons; one enclosed within another.
• Mazes 9 to 11 are an introduction to learning the function that the outer and inner polygons have. The circles in maze 9 hint at something singular, so heading straight for the exit seems likely; maze 10 uses the same principle but also hints at the importance of separation. Maze 11 has the same layout as maze 10 but with a triangle enclosing a cross, so the other route seems to be the correct answer. This is also the first maze that has a different set of polygons, so some property pertaining to these polygons must be significant, potentially the number of sides.
• Mazes 12 and 13 have similar layouts with different polygons, and also have multiple paths to the destination. As the number of sides in each polygon seems to vary, what are some properties those numbers could represent? Maze 12 has four grid squares, and any solution for maze 12 requires an odd number of lines. Five is too high for the number of squares, the number of sides is the next best guess for the exterior polygon. Taking three to be the number of squares enclosed from the way the symbols are presented (five encloses three), there is a unique solution that satisfies these numbers. We can look back at the three previous mazes to see if our proposed rules are true so far. Maze 13 has the same interior polygon but has a even number of sides for the exterior polygon - we have to infer that it is the number of edges bordering the enclosure, as opposed to the number of edges in total. Maze 14 has two polygon sets, and separating the two seems to work the best - combining them is both impossible and has not been alluded to yet. This also indicates that sharing sides for separate polygon sets is inconsequential.
• Maze 15 introduces combining with two polygon sets that appear to be impossible to satisfy separately. The two outer circles force a short solution while the sum of the two inner crosses equals the total number of grid squares. Mazes 16 and 17 revisit the concepts of bordering edges and sharing sides, while maze 18 requires working with two lines while creating equal enclosures.
The solutions to mazes 9-18 can be found here.
• Mazes 19-26: The final mechanic of white squares is introduced.
• Maze 19 indicates that six sides must enclose two squares, which implies a closed loop must exist around the orange polygon set. This is normally impossible, so the white square must contribute to the loop somehow. If we assume that the white square already has a 'pretraced' side, then we can trace a line around the orange polygon set touching both endpoints of the pretraced side, which fulfils the enclosure requirement and has no violations.
• Maze 20 has white squares surrounding the left orange polygon set. If these sides are pretraced then there is no way for three sides to enclose three squares, so there must be a way of undoing the pretracing. Tracing the solution line over these white squares seems to be the most logical method of doing so, such that after applying the effect of the white squares we have the left polygon set satisfied in the same way as the right polygon set. We can conclude here that white squares invert the 'traced state' for the side it is placed in.
• Maze 21 adds crosses back into play after we have deduced the function of white squares. In order for this maze to be completable we have to assume that crosses can be satisfied as long as the solution line passes through it twice, regardless of whatever state the sides surrounding the cross are in after the effect of white squares.
• Maze 22 requires accomodating for two lines when interacting with white squares, in order to fulfil uneven enclosure requirements for each side.
The last four mazes are a recap of every mechanic encountered, and are more of a final test than anything else. The solutions of these eight mazes can be found here, alongside the state of each maze after the white squares are applied for clarity.

Armed with the knowledge of each mechanic we are now able to solve the final maze. This looks fairly intimidating at first glance owing to the large dimensions and the potential for a giant conglomerate of orange polygon sets, but we can make a few observations:

• The broken sides and limited number of squares that are available for enclosure make a giant conglomerate of polygon sets impossible,
• Immediately following that, we can visualise the maze as a series of smaller sub-puzzles, with which our path during or between these sub-puzzles are guided by the rings,
• An cross must have all of its surrounding sides traversed by the solution,
• If left uncombined, a polygon set of four sides enclosing one square implies a closed loop (and variations thereof).
With or without these observations in mind, the solution to the final maze is below:

After applying the white squares, the solution looks like this:

Having completed all the mazes, the question is what can be done with the solutions as a next step. We still haven't used the implicit hint given by maze 2 of finding the shortest path - perhaps counting the length of these paths and translating those to letters may prove fruitful. Additionally, we want to pay attention to the background colour of each maze, of which several particular colours are used throughout and somewhat resemble those of a rainbow. The order of the wires protruding from the final maze are another, more obvious indication of this observation. Grouping the mazes by colour, translating the length of the optimal solutions to a letter and performing a number of fairly simple anagrams yields the following:

Group colourLength of solution (in maze order)Equivalent lettersAnagram
Red19,1,5,3,12SAECLSCALE
Orange4,1,14DANAND
Yellow1,5,11,13AEKMMAKE
Green24,18,15,19XROSXORS
Blue6,15FOOF
Indigo9,2,7IBGBIG
Violet1,7,5,9,13AGEIMIMAGE

This implies scaling the solution of each maze to 10x10 and applying an XOR to the solution of the final maze. We take note of mazes where the grid dimensions are not square or a factor of 10, as these also have coloured wires protruding from certain sides of the maze frame. Indeed, combining the grids of mazes with the same wire colour in the relative positions that the wires indicate create a 5x5 grid that can be scaled - in the example of the mazes with purple wires, the 3x2 maze is placed on the top-right, the 2x5 maze on the left, and the 3x3 maze on the bottom-right (overlapping sides are XOR'ed as normal). Doing the above processes using the solutions of the 'training' mazes (before applying white squares) gives the following XOR:

The abundance of white squares and their apparent lack of purpose in solving the final maze suggests that we apply the above XOR to the post-white-squares solution of the final maze, which gives:

The grid reads "NP POLYTIME VERIFIER", describing the answer CERTIFICATE which, in a more general sense, is otherwise known as a witness. This shares the title of a puzzle game released in 2016 that (arguably) has players discover a verifier to their solutions as much as finding the solutions themselves, and forms the basis to this puzzle.

Design notes:

The aim of the puzzle was to recreate the feeling of going through and discovering the puzzle mechanics within The Witness, while creating an new set of mechanics that preserved as much of the experience in the source material as possible. This also minimised the value or advantage that knowledge of the source material would have, although the first six or so puzzles would be straightforward. With the exception of the white squares, all of the mechanics were devised fairly early on, as was the steps of the overall puzzle. The white squares came late in the puzzle's creation, presenting themselves as a fairly valid and workable puzzle mechanic that was also a sneaky way to make it viable to display a message using the final puzzle's lattice grid. As a side effect the final puzzle ended up looking a little contrived, but trying to create any message without the help of the white squares would require an unbelievable amount of foresight otherwise.

Of course, the way the puzzle was presented meant there was no feedback for solvers as to whether their solutions were correct or optimal. A lot of mazes were reworked in order to smoothen out the learning curve and also to accomodate for changes in the message for the final step, particularly the mazes teaching the function orange polygon sets. Regardless, the learning curve for the white squares ended up being a lot steeper, and the last few training puzzles turned out to be potentially harder than the final puzzle itself, which bumped up the difficulty of what was initially forseen to be three or four stars. One other ambiguity that was overlooked was whether the XOR should be made using the solutions before or after the effect of white squares was applied - it turned out to be to be the former applied to the latter, which was an unnecessary increase in difficulty for a step already prone to mistakes, so apologies for that.

Special thanks to Kevin Liang, for doing a bunch of testing and program-writing to verify the correctness and optimality of every maze!