Putnam Math Competition Lecture Notes

Overview of the Putnam Competition

• A math competition for undergraduate students.
• Duration: 6 hours total, split into two 3-hour sessions.
• Consists of 12 questions.
• Scoring: Each question is scored from 1 to 10; maximum score is 120.
• Notably challenging: Median score is around 1 or 2 despite being taken by math enthusiasts.
• Difficulty increases from questions 1 through 6 in each section.
• Elegant solutions are often found for the hardest problems (questions 5 and 6).

Example Problem: Tetrahedron and Sphere

• Problem: Choose 4 random points on a sphere to form a tetrahedron. Determine the probability that the sphere's center is inside this tetrahedron.
• Approach:
  • Simplify to two dimensions: Consider 3 points on a circle forming a triangle.
  • Determine probability that the triangle contains the circle's center.
  • Fix two points and vary the third. Identify arcs where the triangle contains the center.
  • Probability related to arc size over circle's circumference.
  • Average arc size when choosing random angles is 0.25 of the circle.
  • Extending to 3D is complex but follows a similar logical approach.

Problem-Solving Insights

• Simplifying problems by reducing dimensions.
• Fixing certain elements and varying others to gain insights.
• Using random lines through a center as a conceptual tool.
• Coin-flip analogy for choosing between endpoints of lines; leads to probability calculations.

Generalizing to 3D

• Use random lines through a sphere center.
• Each line has two endpoints; decide with a coin flip.
• 8 possible outcomes for 3 lines; only 1 leads to the desired tetrahedron.
• Solution shows the elegance of mathematical problem-solving.

Educational Takeaways

• Main Lesson: Solve complex problems by breaking them into simpler parts.
• Use additional constructs that simplify understanding.
• Mathematical problem-solving requires intuitive insights and formal articulation.

Additional Puzzle (Sponsored by Brilliant.org)

• Puzzle: 8 students in a circle taking the Putnam; each cheats randomly off a neighbor.
• Find the expected number of students not being cheated off.
• Opportunity to practice problem-solving on Brilliant.org, a platform offering curated math and science problems and courses.
• Promotional offer: Discount available for first 256 sign-ups.

Conclusion

• Importance of developing problem-solving skills.
• Encouragement to engage with mathematical problems through platforms like Brilliant.org to enhance understanding and abilities.