Python/project_euler/problem_116/sol1.py

"""
Project Euler Problem 116: https://projecteuler.net/problem=116

A row of five grey square tiles is to have a number of its tiles
replaced with coloured oblong tiles chosen
from red (length two), green (length three), or blue (length four).

If red tiles are chosen there are exactly seven ways this can be done.

    |red,red|grey|grey|grey|    |grey|red,red|grey|grey|

    |grey|grey|red,red|grey|    |grey|grey|grey|red,red|

    |red,red|red,red|grey|      |red,red|grey|red,red|

    |grey|red,red|red,red|

If green tiles are chosen there are three ways.

    |green,green,green|grey|grey|    |grey|green,green,green|grey|

    |grey|grey|green,green,green|

And if blue tiles are chosen there are two ways.

    |blue,blue,blue,blue|grey|    |grey|blue,blue,blue,blue|

Assuming that colours cannot be mixed there are 7 + 3 + 2 = 12 ways
of replacing the grey tiles in a row measuring five units in length.

How many different ways can the grey tiles in a row measuring fifty units in length
be replaced if colours cannot be mixed and at least one coloured tile must be used?

NOTE: This is related to Problem 117 (https://projecteuler.net/problem=117).
"""


def solution(length: int = 50) -> int:
    """
    Returns the number of different ways can the grey tiles in a row
    of the given length be replaced if colours cannot be mixed
    and at least one coloured tile must be used

    >>> solution(5)
    12
    """

    different_colour_ways_number = [[0] * 3 for _ in range(length + 1)]

    for row_length in range(length + 1):
        for tile_length in range(2, 5):
            for tile_start in range(row_length - tile_length + 1):
                different_colour_ways_number[row_length][tile_length - 2] += (
                    different_colour_ways_number[row_length - tile_start - tile_length][
                        tile_length - 2
                    ]
                    + 1
                )

    return sum(different_colour_ways_number[length])


if __name__ == "__main__":
    print(f"{solution() = }")