Projective Plane Amoeba

This is a puzzle I worked on for a competition in a puzzle Discord server I’m in, which I can also share here now that I’ve been revealed as its author – the theme of the contest was to create a puzzle that ‘is toroidal, recursive, or otherwise wraps around itself somehow such that some or all edges of the grid are spatially connected and continuous’. It is typical for entrants in the contest I was participating in to submit puzzles that are attention-grabbing in their unusualness; I had recently come across this tweet introducing a new genre called Amoeba (or possibly Wamoeba?), and thought there was a charming sort of comedy in the notion of a freehand loop genre, so it seemed the perfect fit.

I also studied just about enough topology during university to know that loops can behave very strangely in different topological spaces, but I didn’t want to make the puzzle completely inaccessible to anybody without a topological background; for this reason, I decided to set a puzzle set on the real projective plane, whose fundamental group is ℤ₂, meaning that there are essentially only two ‘types’ of loops possible in this space; one type is a ‘regular’ loop, i.e., it is contractible to a point and partitions the space into two disconnected components, while the other type of loop goes ‘across’ the entire space once, and does not disconnect the space.

Overall, I’m moderately happy with how the puzzle ended up. I think its solution path is a little dull – lots of trivial deductions, followed by one interesting one at the very end – and it’s also rather overclued, but I’m just proud of myself for actually managing to make a functioning puzzle out of such a bizarre ruleset, especially since I rushed to make it when I came up with the concept a day before the contest deadline. If nothing else, I think I did indeed succeed in creating something attention-grabbingly unusual, and despite only coming in sixth place, the puzzle seemed to be relatively well-received.

Ruleset

In the grid shown, opposite edges have been identified (with matching orientations indicated by arrows) such that the grid is homeomorphic to the real projective plane. Draw a single closed loop (freehand—time to bust out MS Paint!) such that the following conditions are met:

• The loop must not branch or intersect itself.
• The loop must not cross any of the grid’s vertices, or any lines drawn in black.
• The loop must not make any ‘U-turns’, i.e., it may not pass through the same edge of the grid twice consecutively.
• If a grid cell is indicated with a gray number, the loop must pass through that cell as many times as indicated by the number.
• If a grid edge is indicated with a circled number, the loop must pass through that edge as many times as indicated by the number.
• Vertices indicated with a white or black circle must be contained in the interior or exterior (see below) of the loop, respectively.
• The loop must divide the projective plane into two disjoint regions. One region will be homeomorphic to the unit disk (in non-topological terms: if obstacles in the grid were ignored, this region could be smoothly contracted such that it resembles a circle). This region is defined to be the interior of the loop, while the other region (which is not homeomorphic to the unit disk and cannot be contracted to a circle) is the loop’s exterior.