Points, Patterns, and a Mathematical Love Story
In 1933, a young mathematician named Esther Klein noticed something interesting about points on a plane. This observation led to a famous problem – and eventually to her marriage to another mathematician, George Szekeres. Hence, this became known as the “Happy Ending Problem”!
This Week’s Challenge
Basic Concept: A set of points forms a convex shape if you can connect them to form a shape with no dents or indentations.
Example: A triangle is always convex. A star shape is not convex.
Your Tasks
- Starter Question:
Draw 4 points on a piece of paper. Can you always find at least 3 points that form a triangle with no other points inside it?
- Main Challenge:
Draw 5 points in any arrangement (no three points on a straight line). Can you always find 4 points that form a convex quadrilateral (four-sided shape)?
- Investigation:
- Try different arrangements of 5 points
- What makes it easier or harder to find a convex quadrilateral?
- Can you find an arrangement where it’s NOT possible?
Tips for Working on This Problem:
- Start by drawing your points fairly far apart
- Try connecting different sets of 4 points
- Check if any point lies inside the shape you create
- Try to find a systematic way to check all possibilities
Bonus Challenge
How many points do you think you need to be sure of finding a convex pentagon (5-sided shape)? Try to make a conjecture!
Historical Note: This problem led to deep results in combinatorial geometry and inspired many related questions about patterns in point sets. And yes, Esther Klein and George Szekeres got married and continued to work on mathematics together!
