Grasshopper on a triangle

Grasshopper random walk on a triangle — probability after n steps

This is a medium-difficulty probability question testing your ability to model and solve a random walk on a finite graph. It appears regularly in quantitative finance interviews because it rewards both structural insight and careful state-space reasoning.

The core challenge is to set up a recurrence relation or transition-matrix framework that tracks the probability of being at each vertex after each jump. Rather than enumerate all possible paths, successful candidates recognize that symmetry simplifies the problem: only the relative position of the grasshopper matters. From there, you can either build a recurrence for the return probability or use matrix powers to compute the state distribution after a fixed number of steps.

• Markov chains and transition matrices
• Symmetry in state spaces
• Recurrence relations and closed-form formulas
• Parity and periodicity in random walks