Homework - Geometric/Numerical Puzzle

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Problem:

Thirty equally spaced points on the circumference of a circle are labelled in order with the numbers 1 to 30. Which number is diametrically opposite to 7?

Questions:

What process did you use to work on and solve this puzzle?

I created a new form of "degrees" to measure angles (call the units "angle units") where one revolution is equal to 30 angle units. 

Then, if we consider the point labelled 30 as the initial arm of a standard position angle, the point labelled n will correspond to an angle of n angle units. 

Moreover, one half revolution is 30/2 = 15 angle units, and a half revolution will take any point on the circumference of the circle and transform it to its diametrically opposed point. 

Therefore, the number diametrically opposite to 7 is 7 + 15 = 22.  

Could you create other extended puzzles related to this one -- some possible, some impossible?  (Is there any value to giving your students impossible puzzles?)

The following puzzles use the same set-up / configuration as above.

(a) A puzzle possible to solve: How many unique subsets of 3 points from the set of 30 labelled points on the circumference can be selected such that the chords between the points in the subset form an equilateral triangle? Give one example of one such subset, or better yet, provide a general solution for which points should be selected.

(b) Another possible puzzle: How many incongruent triangles can be formed from the chords connecting the set of 30 labelled points?

(c) An impossible puzzle: Suppose a new circle with the same radius is drawn such that the centre of the new circle lies on the point labelled 1. Determine at which of the 30 labelled points the two circles will intersect.

For (c), there is no solution as the circles will intersect at points which are not in the set of 30 labelled points. There is some value to giving impossible problems in that it can help students to generate a deeper understanding of what is possible, to try very creative solution methods, and ultimately work toward proving conjectures that something does not exist. 

What makes a puzzle truly geometric, rather than simply logical? 

I'm not sure how a pure mathematician would answer this question, but I personally think that a geometric puzzle forces us to think spatially in some manner. The trouble with distinguishing the two lies in that there are often multiple representations for problems.

I think I can answer this question better through providing an example though. Extension problem (b) above is very similar to finding the total number of triangles formed by the chords connecting the set of 30 labelled points, which is equivalent to finding the total number of triangles formed by a K-30 graph. Such a problem could be solved without any regard to spatial distribution of the points (it is not intrinsically meaningful to the problem) by simply using combinations (30 choose 3). The geometric aspect comes in the form of the restriction "incongruent triangles". This geometric condition must be interpreted into a rule for selecting subsets in order to continue to apply a combinatorial solution, and so in this case, the spatial distribution does matter. I would say that these two problems side by side show a particular example of the general concept at which I am trying to grasp as the distinguishing factor between a truly geometric problem and one that is simply logical.

Comments

  1. Susan Gerofsky19 October 2020 at 10:03

    Thank you for this very thoughtful and comprehensive post and solution, Adam! Something to consider: how might you word your extension problems to make them easier for, say, Grade 10 students to understand? You've used concise mathematical language here, but your students may not yet be familiar with using that language yet. (And of course, you can help them work towards that proficiency as well!)

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