Shingoki: An Exploration of Unicode

Question

^{An entry in Fortnightly Topic Challenge #42: Wordless Connecting Walls-- wait, what?}

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This is a Shingoki puzzle. The rules (paraphrased from puzzle-shingoki.com) are as follows:

The aim is to form a single loop without crossing or branching paths.

For the loop created to be valid, it must pass through all of the given circles. The circles also have their own rules:

• A line must go straight through a white circle.
• A line must turn 90 degrees upon a black circle.

Lastly, the number in the circle indicates the total length of the lines going out from that circle (disregarding other circles positioned on those lines). In our case, the grid we have has dimensions of 10 units by 10 units.

24-hour hints:

1.

For now I will choose not to say all two groups that have been identified correctly, but just one of them: yes, the final answer looks like a letter 'm'. Keep looking as well; the intended categories are simpler than those speculated by @Deusovi.

2.

Yes, the final answer is a Greek letter. And in fact, knowing these two hints would probably already be sufficient to find the final symbol and then just work backward from there for the remaining categories. Nevertheless, I will say that there is also one aspect characterizing Unicode related to one of the categories.

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_{For convenience. (For the second link, select "Mode" to be "Edge".)}

Answer 1

The Shingoki's solution is:

(More detail is given at the bottom of the post.)

The 16 symbols are:

given by the unused dots (ignoring one of them which is just the default dot in the Penpa editor, as mentioned in the Pastebin link):

Here are the categories:

And there is a character fitting all four of these categories:

The character ϻ, or U+03FB GREEK SMALL LETTER SAN.

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Solving the Shingoki

Initial deductions get us here:

More progress can be made in one of the corners:

The 9 at the bottom must extend at least 5 units upwards. This gives us more progress with the straight 5 near the bottom, and then a column of 2s.

Next, consider

the black 3 near the bottom of the center area. This 3 cannot extend 2 horizontally -- left, it would be blocked by the 2, and right, it would break the 2. So it must extend 2 vertically (and 1 horizontally), and that means it must go up 2.

This leads to some more progress:

Next, look at the 4 clue:

The 4 near the upper left cannot go horizontally, or it would trap the loop above it. This lets us resolve a bunch of other clues, completing the left side of the loop!

And this also lets us resolve the bottom side of the loop:

Here I had to case-bash a bit:

If the top of the 5's segment goes to the right, any possible path either closes the loop too early or strands a segment. (There are three cases: left, up, and right. Right and up break pretty quickly, and left takes a bit more thought to see the break.)

And with this, the puzzle can finally be solved.