Add solution for the Euler project problem 345 (#12666)

* Add solution for the Euler project problem 345.

* Update sol1.py

* Update sol1.py

* Update sol1.py

* Update sol1.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Update sol1.py

* Update sol1.py

* Update sol1.py

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

---------

Co-authored-by: Maxim Smolskiy <mithridatus@mail.ru>
Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

project_euler/problem_345/sol1.py

@@ -0,0 +1,117 @@
+"""
+Project Euler Problem 345: https://projecteuler.net/problem=345
+
+Matrix Sum
+
+We define the Matrix Sum of a matrix as the maximum possible sum of
+matrix elements such that none of the selected elements share the same row or column.
+
+For example, the Matrix Sum of the matrix below equals
+3315 ( = 863 + 383 + 343 + 959 + 767):
+      7  53 183 439 863
+    497 383 563  79 973
+    287  63 343 169 583
+    627 343 773 959 943
+    767 473 103 699 303
+
+Find the Matrix Sum of:
+      7  53 183 439 863 497 383 563  79 973 287  63 343 169 583
+    627 343 773 959 943 767 473 103 699 303 957 703 583 639 913
+    447 283 463  29  23 487 463 993 119 883 327 493 423 159 743
+    217 623   3 399 853 407 103 983  89 463 290 516 212 462 350
+    960 376 682 962 300 780 486 502 912 800 250 346 172 812 350
+    870 456 192 162 593 473 915  45 989 873 823 965 425 329 803
+    973 965 905 919 133 673 665 235 509 613 673 815 165 992 326
+    322 148 972 962 286 255 941 541 265 323 925 281 601  95 973
+    445 721  11 525 473  65 511 164 138 672  18 428 154 448 848
+    414 456 310 312 798 104 566 520 302 248 694 976 430 392 198
+    184 829 373 181 631 101 969 613 840 740 778 458 284 760 390
+    821 461 843 513  17 901 711 993 293 157 274  94 192 156 574
+     34 124   4 878 450 476 712 914 838 669 875 299 823 329 699
+    815 559 813 459 522 788 168 586 966 232 308 833 251 631 107
+    813 883 451 509 615  77 281 613 459 205 380 274 302  35 805
+
+Brute force solution, with caching intermediate steps to speed up the calculation.
+"""
+
+import numpy as np
+from numpy.typing import NDArray
+
+MATRIX_1 = [
+    "7 53 183 439 863",
+    "497 383 563 79 973",
+    "287 63 343 169 583",
+    "627 343 773 959 943",
+    "767 473 103 699 303",
+]
+
+MATRIX_2 = [
+    "7 53 183 439 863 497 383 563 79 973 287 63 343 169 583",
+    "627 343 773 959 943 767 473 103 699 303 957 703 583 639 913",
+    "447 283 463 29 23 487 463 993 119 883 327 493 423 159 743",
+    "217 623 3 399 853 407 103 983 89 463 290 516 212 462 350",
+    "960 376 682 962 300 780 486 502 912 800 250 346 172 812 350",
+    "870 456 192 162 593 473 915 45 989 873 823 965 425 329 803",
+    "973 965 905 919 133 673 665 235 509 613 673 815 165 992 326",
+    "322 148 972 962 286 255 941 541 265 323 925 281 601 95 973",
+    "445 721 11 525 473 65 511 164 138 672 18 428 154 448 848",
+    "414 456 310 312 798 104 566 520 302 248 694 976 430 392 198",
+    "184 829 373 181 631 101 969 613 840 740 778 458 284 760 390",
+    "821 461 843 513 17 901 711 993 293 157 274 94 192 156 574",
+    "34 124 4 878 450 476 712 914 838 669 875 299 823 329 699",
+    "815 559 813 459 522 788 168 586 966 232 308 833 251 631 107",
+    "813 883 451 509 615 77 281 613 459 205 380 274 302 35 805",
+]
+
+
+def solve(arr: NDArray, row: int, cols: set[int], cache: dict[str, int]) -> int:
+    """
+    Finds the max sum for array `arr` starting with row index `row`, and with columns
+    included in `cols`. `cache` is used for caching intermediate results.
+
+    >>> solve(arr=np.array([[1, 2], [3, 4]]), row=0, cols={0, 1}, cache={})
+    5
+    """
+
+    cache_id = f"{row}, {sorted(cols)}"
+    if cache_id in cache:
+        return cache[cache_id]
+
+    if row == len(arr):
+        return 0
+
+    max_sum = 0
+    for col in cols:
+        new_cols = cols - {col}
+        max_sum = max(
+            max_sum,
+            int(arr[row, col])
+            + solve(arr=arr, row=row + 1, cols=new_cols, cache=cache),
+        )
+    cache[cache_id] = max_sum
+    return max_sum
+
+
+def solution(matrix_str: list[str] = MATRIX_2) -> int:
+    """
+    Takes list of strings `matrix_str` to parse the matrix and calculates the max sum.
+
+    >>> solution(["1 2", "3 4"])
+    5
+    >>> solution(MATRIX_1)
+    3315
+    """
+
+    n = len(matrix_str)
+    arr = np.empty(shape=(n, n), dtype=int)
+    for row, matrix_row_str in enumerate(matrix_str):
+        matrix_row_list_str = matrix_row_str.split()
+        for col, elem_str in enumerate(matrix_row_list_str):
+            arr[row, col] = int(elem_str)
+
+    cache: dict[str, int] = {}
+    return solve(arr=arr, row=0, cols=set(range(n)), cache=cache)
+
+
+if __name__ == "__main__":
+    print(f"{solution() = }")