Games 1 and 2: Dealing with Diagonals
Challenge 1
Two teams (A and B) of \(n\) players each are seated around a \(2\cdot n\)-gon polygonal table. Team A are allocated the odd-numbered seats, and team B, the even numbers.
How do we set up the draw so that each member of team A competes against each member of team B exactly once?
How many times must they meet?
Challenge 2
Consider a clock face with \(2\cdot n\) "hours" marked around the dial.
Let \(a(n) = \) number of ways to match the even hours to the odd hours, modulo rotations and reflections.
OR... Given a \(2\cdot n\)-gon... Draw \(n\) diagonals so that they don't share end points, and there are an odd number of polygon sides between the end-points.
How many such variants do you think can be found for a hexagon (\(n=3\))? For an octagon (\(n=4\)), decagon (\(n=5\)) - or the standard clock dodecagon (\(n=6\))? And why stop there - try \(n=7, 8, 9, 10\).
Below you will find \(n\) draggable segments. Use these to find as many solutions (variants) for these challenges as you can. (Vary the number of sides using the \(n\) controls and browse some sample solutions using \(Variants\))
(NOTE: To model Challenge 2, use negative values of \(v\)).
To begin, you might try to:
drag the segments to build three different variants for the hexagon.
Press Check to check your models.
drag the segments to build a few different variants for the octagon.
Press Check to check your models.
drag the segments to build some different variants for the decagon.
Press Check to check your models.
Try a few more, browse the samples, and then try the Quiz!
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