Geometric Puzzle

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The first thought that came to my mind when I saw this problem was circumference. Since the points are evenly spaced apart, I could divide the circumference by the total amount of points and figure out the distance between each point. Then I thought I could scale the problem down to less points, so I thought about a clock. A clock has 12 numbers that are all evenly spaced apart. So diametrically opposite of 12 is clearly 6, the diametrical opposite of 1 is clearly 7. The pattern would be just adding 6 to your number to find the diametrical number. This pattern comes from dividing the total number of points by 2. This makes sense because numbers that are diametrically opposite from each other on a circle, must be spaced half-way around the circumference. Since we have 30 points, to find numbers diametrically opposite, we have to add 30/2 = 15. So for 7, if we add 15 we get 22; diametrically opposite of 7 is 22. You could also imagine if you unwinded the circle into a straight line the total distance would be 2pi*r. Then points that are diametrically opposite should have a distance of pi*r away from each other.

You could extend this problem by asking another question such as, what 2 pairs of diametrically opposite numbers would form lines that are perpendicular?. For example in our case, you could ask, "Since 7 and 22 are diametrically opposite, what other diametric opposite pair would form a perpendicular line (with 7 and 22)". For this problem, if you were to use an odd number of points, I believe it would be impossible. This is because if you have an odd number of points around a circle it would not be symmetrical, which means there would no diametrically opposite numbers. I think there is value in giving impossible problems, but I don't feel like the teacher should be 'tricking' the student. The teacher shouldn't present it as a problem with a clear solution, they could say something like, "This one is challenging so make sure to explain your answer and see if it makes sense!". I think these impossible problems would work better as a challenge for students who are 'stronger'. There are some students who excel by solving problems using a concrete recipe. By throwing these curveballs at them, they learn how to understand when they can apply these concepts and why sometimes they don't always work.

I think the difference between geometrical and logical puzzles would have to involve the knowledge of how to understand and apply geometric facts to get the solution.You could use logic to get your answer, but you would be able to use geometric facts to prove your answer. For this problem I started off using logic to predict the answer, then I used geometric facts to solidify and prove my predictions.

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