A case study in problem difficulty (Score 2)

As a professional mathematician specializing in representation theory, I evaluated these problems. One problem in particular caught my attention: "Orbit Counting of Matrix Tuples" (https://epoch.ai/frontiermath/benchmark-problems), rated as Medium-Low difficulty. I believe this problem was constructed "backwards" – taking a standard problem about classifying representations of the Coxeter group for a given Dynkin diagram and deliberately obscuring it to appear more intimidating. Once you decode its true nature, the solution becomes straightforward and can be implemented with a simple Python script. The key technical challenge lies in identifying the correct Dynkin diagram, which depends on finding pairs (i,j) where 3j-i is a positive multiple of 5. In fact, I initially missed the crucial "positive" qualifier and derived an incorrect graph and solution. Credit to the problem designers for this subtle trap! However, there's a significant issue with the official solution. While it identifies the Dynkin graph as a path of length 4 and claims correspondence with the symmetric group S_5's Dynkin diagram, this is incorrect. The relation defined by the (i,j) pairs isn't symmetric – when you interchange i and j, you get generators that paradoxically satisfy both order-3 and order-2 (commutative) Coxeter relations. Following this through logically, all generators must be equal, resulting in nothing more exotic than a cyclic group of order 2. This group has exactly two irreducible representations, both of degree 1, yielding an answer of 1001. While this is the correct solution, its simplicity is almost misleading – someone could stumble upon it without truly understanding the underlying mathematics. Suggestion for improvement: The problem could be strengthened by properly defining a symmetric relation for the Dynkin graph. With this modification, it would serve as an excellent test of basic representation theory understanding. In summary: +1 for the clever "positive multiple" trap, -1 for the flawed graph definition that oversimplifies the solution.