Geometric/numerical puzzle

 The answer is 22. Since there are thirty numbers around the circle, and the number diametrically opposite a given number is halfway around from that number, it must be 15 places away. Therefore, the number diametrically opposite 7 is 7+15 = 22. We can draw a diagram to confirm this. Alternatively, we could have drawn a diagram with a smaller number of points placed around the circle.

Here's a related problem with a solution:

1. Some number of points are equally spaced on the circumference of a circle, and labeled with increasing numbers. If 3 and 32 are diametrially opposite, how many points are on the circle?

And a couple without solutions:

2. 45 points are placed equally distant around the circumference of a circle, and labeled in order with the numbers 1 to 45. What number is diametrically opposed to the number 16?

3. Sarah stands on a circle with radius 5 meters, and puts the number 1 where she stands. She then walks one meter along the circle clockwise and places the number 2, one more meter along the circle and places the number 3, and so on. She decides she will stop when she place a number exactly on top of another number. What number does she stop on?

This third problem is quite difficult, and is more of a question about irrational numbers than geometry or number theory, but I think it's an interesting one for getting students to think about irrationality.

Problems without solutions are very valuable - they encourage students to try many different ideas to solve them, because the first approach they try will inevitably fail. They also prompt the student to make an argument for why the problem has no solutions, which is difficult and very helpful in building a good understanding of the problem. It's possible to solve a problem without fully understanding it, but it's not possible to show that it has no solutions without fully understanding it.

I would argue that a puzzle isn't truly geometric unless the simplest, or most natural, solution is one that involves geometry - that is, some sort of concept of shape or space. Many geometric problems can, with difficulty, be reduced to a purely logical argument, but that usually makes them harder. Some logical problems can be reduced to geometric arguments, and this is often easier than solving them with logic.