Geometric Puzzle

What process did you use to work on and solve this puzzle?
This puzzle came a lot more naturally to me than the other before. Since there are an even number of points, each point would line up with another on the other side. Knowing this, I found the angle each segment would be and found that from the first point (1) to 7 is 84 degrees. Since we want the point exactly opposite, this point would be 180 degrees from 7. Doing some simple math I found the answer to be 22. I drew a picture to confirm my answer but it was not executed well (my compass broke) and it was difficult to follow the lines to their counterparts. If the circle had been perfectly drawn, it would have been easier to see. Also, if the number of points was smaller, say 6, one could easily find the diametric by simply tracing the lines back to the number.

Could you create other extended puzzles related to this one -- some possible, some impossible?  (Is there any value to giving your students impossible puzzles?)
This problem could easily be turned impossible by making the number of equally spaced points an odd number. Along the lines of geometry, you could present some ancient problems to students such as cube duplication and angle trisection. After students have struggled for a bit, you can tell them how there is no answer and people have worked for years and still have not been able to find a solution. I think there is some value in this because students become discouraged when they cannot figure out the answer. Learning that many mathematicians were unable to find solutions to these problems could be reassuring that just because an answer cannot be found, does not mean you cannot do math. Furthermore, I would bring up the Millennium Prize Problems and other unsolved problems in mathematics. Having some problems that are possible will also be worth exploring in class. In terms of geometry, I would try to bring in some other cultures geometric problems to get students practicing math while also learning some history. For example, bringing problems about volume of irregular shapes, approximations of pi and the Pythagorean Theorem from Ancient Babylonia, Ancient China and Ancient Indian.

What makes a puzzle truly geometric rather than simply logical?
This problem is purely geometrical because one does not need to do any computation. Knowing the properties of a circle and being able to see is a way to solve it. Once the diagram is set up, all one has to do is trace the line from 7 through the center and to the other side which will end up at 22. Problems were students do not need to compute and set up equations and do algebra to solve a problem, helps make it geometric. Although the problems could be solved logically, that is not the only way and typically the geometric way is easier.