In my process of solving, I first tried drawing simply cases of 2 points, 3 points etc. so I could visualize the problem. Then, I realized that the point directly opposite the last one was half, so for the 30 case, all I had to do was write 1-15, then 16-30 underneath it and find the pair to 7, which is 22. Generally, this would be T/2 + N for N =< T/2, and N - T/2 for N > T/2, where T is the total number of points, and N is the number you're trying to match with its partner.
I tried to extend this with a square: A square is drawn with 32 points (numbered clockwise) spread out so that each side's points are equally distributed across the line. If the square was folded so that point 1 and 17 overlapped, what point would overlap with 3?
The solution I got was 15, but I wonder if there would be misinterpretations. That's why I said the square was folded in half in the first place - if I tried to say the point across a given point, there could be a lot of misinterpretation. Also, the numbers on the vertices would have multiple answers (it could be 1 (across the diagonal), 2 (the vertices on either end of 90 degrees), or more because "across" isn't very specific).
When trying to make an impossible answer, I chose the circle with 3 points and asking what number was across 3. None of these points should have a partner across their diagonal so there would be no solution. I honestly don't like impossible puzzles unless the student knows in advance that that is a possibility (e.g. when solving system of equations, students are prepared for and can confidently check if there is no solution). The benefit of impossible puzzles is that it builds confidence and deeper understanding in the students to be able to stand by their answer and explain why it is impossible.
I would say a question is geometric if you're solving for a geometric quantity (e.g. measurements, angles etc.). In this case, even though there are geometric shapes and we're solving for the numbered point, it really has more to do with the organization of numbers than the shape itself.
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