```python
from typing import Dict

def cut_roll(length: int, board: list[tuple[int, int]]) -> Dict[str, any]:
    if length < 0:
        raise ValueError("Length must be a non-negative integer.")

    def max_takeable_piece(index: int) -> int:
        """Return the maximum length of a piece that can be cut from the roll."""
        return min(length - index, board[index][0])

    def max_tookable_price(index: int) -> int:
        """Return the price of the most valuable piece that can be cut from the roll."""
        return board[index][1] if index < len(board) else 0

    def dfs(index: int, current_length: int, current_piecelength: int) -> Tuple[int, list]:
        """
        Recursively find the maximum price and pieces for a given length of roll.
        :param index: Current board piece to consider.
        :param current_length: Length of the remaining roll.
        :param current_piecelength: Length of the last piece cut.
        """
        if current_length == 0:
            return (0, [])

        # Try cutting the current board piece
        max_piece = max_takeable_piece(index)
        if max_piece > 0:
            take_price = board[index][1]
            take_pieces = dfs(index + 1, current_length - max_piece, max_piece) + [max_piece]

            # Try scrapping the last metre of the roll
            scrap_price = 0
            scrap_pieces = dfs(index + 1, current_length - 1, current_piecelength)

            # Choose the option with the highest price or, in case of a tie, longest piece
            return max((take_price, take_pieces), (scrap_price, scrap_pieces))

        # If no more board pieces fit, return the best price from scraps
        return dfs(index + 1, current_length - 1, current_piecelength)

    result = dfs(0, length, 0)
    return {
        "takings": result[0],
        "greatest_price": max_tookable_price(board.index(max_tookable_piece(0))),
        "pieces": result[1][::-1]
    }
```