Geometric / Numeric Puzzle

22 Would be diametrically opposite to 7. The process that I used to work on this problem was to relate it to a similar situation, a clock. This question can be thought of as a clock that goes to 30 instead of 12. On this special new 30-clock, every number is bigger than on a regular clock by a factor of 30/12 or 5/2 or 2.5. So 7 on this clock would be 14/5 on a regular clock, which is 2.8 or just before 3 o’clock. Thinking about the familiar clock, we can realize that diametrically opposite to just before 3 o’clock, is just before 9 o’clock. The distance before 3 would be the same as the distance before 9. To be exact, it is 8.8 or 44/5 so on our new 30-clock, 44/5 would be 44/5 * 5/2 which is 22. So, 22 is diametrically opposed to 7 on this 30 point circle.

                I’m torn on the issue of impossible questions because on the one hand, they may inspire students, but on the other hand, they may repel students. While I don’t doubt that a small number of students who are very curious about math may find some joy and valuable contemplation in impossible questions, I believe that most students would be disheartened by the prospect of an impossible question and overall, impossible questions may turn more students off of math than they would bring students towards it. There are many other ways to inspire curious math students. Therefore, this approach of impossible questions should probably be foregone or restricted to only the students interested in them, as it may result in the exclusion of some students.

                When I think of geometry, I think of shapes. So what makes a puzzle geometric (instead of simply logical) is the involvement of shapes as a necessary aspect of the puzzle. That being said, a geometric puzzle can often be solved by other means. Similarly, a non-geometric puzzle can sometimes be solved geometrically.