Solution to the Psychoanalytix puzzle

Psychoanalytix is one of the interesting puzzles of the Melbourne University Puzzle Hunt 2013 competition. The plot of the game that year was based on the characters from the Asterix & Obelix comics, and each of its puzzles was related to one of the inhabitants of the Gaul village or to one of the Romans. Psychoanalytix is ​​one of the villagers, his puzzle was part of the fifth act of the game. The puzzle was preceded by an image of this character, as well as a brief description of him: "A trailblazing druid, Psychoanalytix determined that it is the interference between left and right hemispheres of the brain that plays a critical role in the formation of thoughts."

The puzzle consisted of a 12x10 grid. In the lower left corner was the S square (start); in the upper right corner was the F square (finish). Some squares contained numbers from 0 to 5. Additionally, the grid contained five line fragments, each occupying two squares.

The puzzle was preceded by a hint that shed light on the rules for filling the grid.

Its center was occupied by the brain's hemispheres. Near the left hemisphere was a small grid called LEFT SIDE, and near the right hemisphere was an identical grid called RIGHT SIDE. It had a size of 4x4 and, like the larger grid, contained S and F squares in the same positions, as well as numbers in some squares, from 0 to 4. Each small grid, however, had its own path from S to F. It turns out that these two paths were constructed according to different interpretations of the grid's numbers. For the left side the numbers indicated how many times the path has a corner segment in that square or in the adjacent squares. For the right side the numbers indicated how many times the path has a straight segment in that square or in the adjacent squares.

Thus, the task asked to build a path in the large grid from the S square to the F square in two different ways, according to the rules for the left and right sides.

The author of this interesting puzzle is Andrew Elvey Price, who in 2013 was Vice-President and then President of the Melbourne University Mathematics and Statistics Society (he is also the author of the Informatix puzzle).

Below are examples of constructing each of the two paths. When solving the puzzle, it can be helpful to mark the squares that must contain a corner/straight segment and those that cannot contain a corner/straight segment. In some cases it's necessary to consider all possible path continuations in a given area of ​​the grid to eliminate alternatives that contradict the puzzle's rules.

Left Side

Right Side

Now that the solutions have been found, all that remains is to convert them into the answer to the task. The description of Psychoanalytix along with the hint suggests that both solutions should be somehow combined. One could try using logical operations for this purpose. If we XOR together the found paths we will obtain the following pattern.

The fragments of the paths form the word DUALITY, which is the answer to the puzzle.

In the solution's note Andrew Elvey Price stated that "Original credit for these puzzles goes to Erich Friedman (here and here)."