N Queens Backtracking in C#

using System;
// N Queens Backtracking in C#
//
// PURPOSE
// N-queens places one queen in each row with no shared columns or diagonals.
//
// METHOD
// Try columns row by row. Keep occupied-column and diagonal marker arrays for constant-time
// safety checks. Backtrack when no column remains and count complete placements.
//
// COMPLEXITY
// Exponential search, bounded by O(n!) candidate placements with additional column-scan
// overhead; O(n) auxiliary space.
//
// CONTRACT
// 0<=n<=12. Returns solution count. The empty board has one placement. This implementation
// avoids recursion.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// C# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 1.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class Program {
    static void alg_require(bool valid) { if (!valid) throw new ArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int n_queens(int n) {
        checked {
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((n >= 0) && (n <= 12)));
            int[] columns = new int[alg_size((n + 1))];
            int[] rising = new int[alg_size(((2 * n) + 1))];
            int[] falling = new int[alg_size(((2 * n) + 1))];
            int[] placement = new int[alg_size((n + 1))];
            int row = 0;
            int col = 0;
            int count = 0;
            if ((n == 0)) {
                return 1;
            }
            while ((row >= 0)) {
                col = placement[row];
                if ((col > 0)) {
                    columns[(col - 1)] = 0;
                    rising[((row + col) - 1)] = 0;
                    falling[((row - col) + n)] = 0;
                }
                while (((col < n) && ((columns[col] != 0) || (rising[(row + col)] != 0) || (falling[(((row - col) + n) - 1)] != 0)))) {
                    col = (col + 1);
                }
                if ((col == n)) {
                    placement[row] = 0;
                    row = (row - 1);
                } else {
                    placement[row] = (col + 1);
                    columns[col] = 1;
                    rising[(row + col)] = 1;
                    falling[(((row - col) + n) - 1)] = 1;
                    if ((row == (n - 1))) {
                        count = (count + 1);
                    } else {
                        row = (row + 1);
                        placement[row] = 0;
                    }
                }
            }
            return count;
        }
    }

    static void Main() {
        int n = 0;
        int result = n_queens(n);
        Console.WriteLine(result);
    }
}