#### Minimax

```"""
Minimax helps to achieve maximum score in a game by checking all possible moves
depth is current depth in game tree.

nodeIndex is index of current node in scores[].
if move is of maximizer return true else false
leaves of game tree is stored in scores[]
height is maximum height of Game tree
"""
from __future__ import annotations

import math

def minimax(
depth: int, node_index: int, is_max: bool, scores: list[int], height: float
) -> int:
"""
This function implements the minimax algorithm, which helps achieve the optimal
score for a player in a two-player game by checking all possible moves.
If the player is the maximizer, then the score is maximized.
If the player is the minimizer, then the score is minimized.

Parameters:
- depth: Current depth in the game tree.
- node_index: Index of the current node in the scores list.
- is_max: A boolean indicating whether the current move
is for the maximizer (True) or minimizer (False).
- scores: A list containing the scores of the leaves of the game tree.
- height: The maximum height of the game tree.

Returns:
- An integer representing the optimal score for the current player.

>>> import math
>>> 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 len(scores) == 0:
raise ValueError("Scores cannot be empty")

# Base case: If the current depth equals the height of the tree,
# return the score of the current node.
if depth == height:
return scores[node_index]

# If it's the maximizer's turn, choose the maximum score
# between the two possible moves.
if is_max:
return max(
minimax(depth + 1, node_index * 2, False, scores, height),
minimax(depth + 1, node_index * 2 + 1, False, scores, height),
)

# If it's the minimizer's turn, choose the minimum score
# between the two possible moves.
return min(
minimax(depth + 1, node_index * 2, True, scores, height),
minimax(depth + 1, node_index * 2 + 1, True, scores, height),
)

def main() -> None:
# Sample scores and height calculation
scores = [90, 23, 6, 33, 21, 65, 123, 34423]
height = math.log(len(scores), 2)

# Calculate and print the optimal value using the minimax algorithm
print("Optimal value : ", end="")
print(minimax(0, 0, True, scores, height))

if __name__ == "__main__":
import doctest

doctest.testmod()
main()
```  