Points and Triangles on a Circle
Added at archival (these are more puzzle-like):
N points are arranged uniformly around a circle. How many triangles can
be formed which have three of these points as vertices?
Two players in turn draw lines between two of the points on the above
circle, using red and blue marks respectively. The loser is the
first to create a triangle in his/her color, with vertices on three of
the N points.
Is there a strategy to this game? How can one person play to win (or
not to lose)?
Can you analyze the game if the created triangle does not
need all three vertices on the circle? (Vertices
could be found within the circle if same-color lines cross.)
If you want to focus, I'd be interested in numerical examples for N=4,5,6,7.
- Can the above game end in a draw for any N? That is, can all
possible lines be red or blue, but have no triangles of one color with
vertices on the circle? (Ideally, the number of red and blue lines would
differ by at most 1, but I'd be interested in solutions in which they
differ by more.)
- What is the largest number of lines which can be drawn connecting the
N points and not have a triangle with vertices on the circle?
(Just a single color for this puzzle. To label your solution, it may be
easiest to number the points 1-to-N, then just list the endpoints of the
lines you didn't draw.)
Source: Original, based on a game on a breakfast cereal box.
Mail to Ken
My birthday was July 27! I finally reached the first 2-digit
(How's that for a puzzler's birthday?)