Problem of the Week #48: Monday December 4th, 2023 As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
This problem is inspired by this twitter post.
The original problem starts with a polygon and a number of points inside it. We are asked to draw line segments between the points to subdivide the entire polygon into triangles such that
– no line segment goes through a point except at the end points,
– no point inside the polygon is left unused
– the resulting graph of triangles can be coloured using two colours such that any two triangles sharing a side must have different colours
The following figure illustrates dividing a heptagon with 3 inner points into triangles. The left one can be coloured using two colours. The right one cannot.
a). Find one way to subdivide the polygon that satisfies the conditions. How many sides is your polygon? How many inner points did you use?
b). Is it possible to divide a triangle with 4 inner points this way?
c). Is it possible for a quadrilateral with 3 inner points?
d). For which numbers of inner points is it possible inside a quadrilateral?
e). For which polygons is it possible with 2 inner points?
f). How can we determine if this is possible for an n-gon with k inner points, where n and k are natural numbers?
g). Would your answer to part f) be different if we also include the condition that no segment can be made between to points on the polygon?
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.