Update 2015 solutions

lib.py

@@ -226,7 +226,7 @@ def count_trailing_repeats(lst):
 class A_Star(object):
     def __init__(self, starts, is_goal, h, d, neighbors):
         """
-        :param h: heuristic function
+        :param h: heuristic function (never overestimate)
         :param d: cost from node to node function
         :param neighbors: neighbors function
         """
@@ -297,6 +297,35 @@ def mod_inverse(a, m):
 
     g, x, _ = egcd(a, m)
     if g != 1:
-        raise Exception('Modular inverse does not exist')
+        raise Exception("Modular inverse does not exist")
     else:
         return x % m
+
+
+class V:
+    def __init__(self, *args):
+        self.xs: tuple[int] = tuple(args)
+
+    def __eq__(self, other):
+        if isinstance(other, V):
+            return all(v1 == v2 for v1, v2 in zip(self.xs, other.xs))
+        elif hasattr(other, "__len__") and len(self.xs) == len(other):
+            return all(v1 == v2 for v1, v2 in zip(self.xs, other))
+        return False
+
+    def __getitem__(self, i: int):
+        return self.xs[i]
+
+    def __hash__(self):
+        return hash(self.xs)
+
+    def __add__(self, other):
+        if isinstance(other, V):
+            return V(*[v1 + v2 for v1, v2 in zip(self.xs, other.xs)])
+        assert hasattr(other, "__len__"), f"V.__add__({self}, {other}) missing `len`"
+        assert len(self.xs) == len(other), f"V.__add__({self}, {other}) `len` mismatch"
+        return V(*[v1 + v2 for v1, v2 in zip(self.xs, other)])
+
+    def __repr__(self):
+        s = ", ".join(map(str, self.xs))
+        return f"V({s})"