Ciro Santilli @cirosantilli 40

Required Characteristics

PhD-level difficulty: Suitable for qualifying exams, research papers, or advanced seminars

Requires genuine insight: Not solvable by routine application of known algorithms

Clear proof-based main question: Answer should be a complete mathematical argument, not just a number

2-3 unique-answer subquestions: Enable automated evaluation (e.g., "Is the statement true for n=5?", "What is the rank of this group?")

Example 1: Stable Graphs

Main question: Find a closed formula for the number $N(g)$ of stable graphs of genus $g$ with no legs and precisely 3 edges, for all $g\ge 2$.

Subquestions:

• What is $N(3)$?
• What is $N(8)$?
• What is $N(1000)$?