Add Project Euler problem 116 solution 1 (#6305)

* Add solution

* updating DIRECTORY.md

* Fix pre-commit

* updating DIRECTORY.md

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: John Law <johnlaw.po@gmail.com>

project_euler/problem_116/sol1.py

@@ -0,0 +1,64 @@
+"""
+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() = }")