Distinguish Endpoint and Survival Probabilities in a 3D Diagonal Walk

A walk starts at an integer position (x,y,z) in an8x8x8 grid, with0<=x,y,z<8. On each step it independently chooses one of the eight vectors (±1,±1,±1), each with probability1/8. Let0<=N<20. The reported wording asks about being inside after exactly N steps, while its described DP discards moves outside the grid at every step. Explain the distinction and solve both interpretations rather than treating them as equivalent. ### Part 1 — Final Position Only Find the probability that the position after N steps is inside the grid, allowing a path to leave and later return. #### What This Part Should Cover Retain returning paths and exploit the independence of coordinate increments if useful. ### Part 2 — Stay Inside Throughout Find the probability that the walk never leaves the grid during its N steps. #### What This Part Should Cover Use a grid-state probability DP with outside transitions discarded and explain the meaning of lost probability mass. ### What a Strong Answer Covers A correct formula or recurrence for each interpretation, an example showing they differ, and time and space complexity. The missing boundary policy must be resolved before creating a single-output coding contract. ### Follow-up Questions - What does each interpretation return for N=0? - Why must surviving transitions retain probability1/8 rather than be renormalized over legal moves?