Add solution for the Euler project problem 164. (#12663)

* Add solution for the Euler project problem 164.

* Update sol1.py

* Update sol1.py

* Update sol1.py

* Update sol1.py

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* Update sol1.py

---------

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

Author: 3 people

project_euler/problem_164/__init__.py

@@ -0,0 +1,65 @@
+"""
+Project Euler Problem 164: https://projecteuler.net/problem=164
+
+Three Consecutive Digital Sum Limit
+
+How many 20 digit numbers n (without any leading zero) exist such that no three
+consecutive digits of n have a sum greater than 9?
+
+Brute-force recursive solution with caching of intermediate results.
+"""
+
+
+def solve(
+    digit: int, prev1: int, prev2: int, sum_max: int, first: bool, cache: dict[str, int]
+) -> int:
+    """
+    Solve for remaining 'digit' digits, with previous 'prev1' digit, and
+    previous-previous 'prev2' digit, total sum of 'sum_max'.
+    Pass around 'cache' to store/reuse intermediate results.
+
+    >>> solve(digit=1, prev1=0, prev2=0, sum_max=9, first=True, cache={})
+    9
+    >>> solve(digit=1, prev1=0, prev2=0, sum_max=9, first=False, cache={})
+    10
+    """
+    if digit == 0:
+        return 1
+
+    cache_str = f"{digit},{prev1},{prev2}"
+    if cache_str in cache:
+        return cache[cache_str]
+
+    comb = 0
+    for curr in range(sum_max - prev1 - prev2 + 1):
+        if first and curr == 0:
+            continue
+
+        comb += solve(
+            digit=digit - 1,
+            prev1=curr,
+            prev2=prev1,
+            sum_max=sum_max,
+            first=False,
+            cache=cache,
+        )
+
+    cache[cache_str] = comb
+    return comb
+
+
+def solution(n_digits: int = 20) -> int:
+    """
+    Solves the problem for n_digits number of digits.
+
+    >>> solution(2)
+    45
+    >>> solution(10)
+    21838806
+    """
+    cache: dict[str, int] = {}
+    return solve(digit=n_digits, prev1=0, prev2=0, sum_max=9, first=True, cache=cache)
+
+
+if __name__ == "__main__":
+    print(f"{solution(10) = }")