Diametric Opposites Puzzle

With 30 equally spaced points on the circumference of a circle, which number is diametrically opposed to 7?

My general approach with puzzles is to decompose the problem to a simple case which I can draw and validate easily. If we let n be the number of points equally spaced on the circumference of a circle and start with n = 1, 2, 3, 4, 5 ... we can see that odd n cases do not have diametric points. So, we must only consider even n cases. With n = 2, 4, 6, I noticed that the diametric pairs, denoted (a, b), were respectively (1, 2); (1, 3), (2, 4); and (1, 4), (2, 5), (3,6). From these observations I hypothesized that for a ≤ n/2, b = a + n/2. I then tested this hypothesis for n = 12, which is just like a clock. The six diametric pairs matched the formula. With n = 30, and a = 7, b = 7 + 30/2 = 22. Thus, 22 is diametrically opposed to 7. 

To solve this puzzle I used a base case approach like in proofs by induction. I verified my work by comparing hypothesized relationships with geometric representations on paper. If I were to extend this puzzle, I would give my students an n very large such that they wouldn't be able to draw it out easily, much like the idea of our first puzzle with 1000 lockers. I might consider 366, the number of days in a leap year. I could also extend this problem to ask students to find the set of three points which form an equilateral triangle, since 366 is also divisible by 3.

With regards to impossible puzzles,  I am hesitant to put my students to such a task. Though the learning process of facing the problem is what is important, I would not want to have students distrust my future problems and puzzles where they might think it is impossible if they cannot find the solution. I do like the idea of providing difficult puzzles or challenging bonus questions which are seemingly impossible, so that I may reward the thought process of students as they approach the problem. In class we could work through the puzzle or problem together as a group to get to the solution through an inquiry approach where I guide the discovery of the solution instead of just telling them. 

Lastly, in terms of puzzles that are truly geometric versus simply logical, I'm not quite sure how to discern the distinction. I think every puzzle involves logic and that every puzzle can have a physical representation. However the physical representation might not be truly geometric, in that it may serve as a visual tool to solve the problem.