#### Minmax

M
```"""
Minimax helps to achieve maximum score in a game by checking all possible moves.

"""
from __future__ import annotations

import math

def minimax(
depth: int, node_index: int, is_max: bool, scores: list[int], height: float
) -> int:
"""
depth is current depth in game tree.
node_index is index of current node in scores[].
scores[] contains the leaves of game tree.
height is maximum height of game tree.

>>> scores = [90, 23, 6, 33, 21, 65, 123, 34423]
>>> height = math.log(len(scores), 2)
>>> minimax(0, 0, True, scores, height)
65
>>> minimax(-1, 0, True, scores, height)
Traceback (most recent call last):
...
ValueError: Depth cannot be less than 0
>>> minimax(0, 0, True, [], 2)
Traceback (most recent call last):
...
ValueError: Scores cannot be empty
>>> scores = [3, 5, 2, 9, 12, 5, 23, 23]
>>> height = math.log(len(scores), 2)
>>> minimax(0, 0, True, scores, height)
12
"""

if depth < 0:
raise ValueError("Depth cannot be less than 0")

if not scores:
raise ValueError("Scores cannot be empty")

if depth == height:
return scores[node_index]

return (
max(
minimax(depth + 1, node_index * 2, False, scores, height),
minimax(depth + 1, node_index * 2 + 1, False, scores, height),
)
if is_max
else min(
minimax(depth + 1, node_index * 2, True, scores, height),
minimax(depth + 1, node_index * 2 + 1, True, scores, height),
)
)

def main() -> None:
scores = [90, 23, 6, 33, 21, 65, 123, 34423]
height = math.log(len(scores), 2)
print(f"Optimal value : {minimax(0, 0, True, scores, height)}")

if __name__ == "__main__":
import doctest

doctest.testmod()
main()
```