Challenge: N-Queens

Start with the idea

Solve the exact task, not just the visible example: Return the number of solutions for a 5x5 board. On input n=5, predict the output before opening the answer reveal. The revealed output is a test oracle for this case, while private checks exercise other legal cases. Start by stating the input grammar and output format in your own words. Parse the complete input, including surrounding whitespace, before relying on any convenient positions in the example.

Build the solution around one maintained claim: Place one queen per row and reject occupied columns and both diagonals before recursing. Decide what state represents that claim, how it is initialized, what a single update does, and when the algorithm stops. Most patterns here explore a decision tree; meet-in-the-middle instead enumerates two halves and combines them. Define the partial choice or subset summary that the pattern retains. Trace the visible input by hand and record at least two intermediate states, not only the final answer. If the trace cannot justify your predicted output, there is a gap between the idea and the implementation.

Use the Rust sketch as a guide, not a substitute for a complete solve function. The supplied Rust sketch is a starting point. Identify what it stores, how one iteration changes it, and whether it already solves the exact task or only demonstrates a related operation.