Update 2015 solutions

2015/d1.py

@@ -1,8 +1,9 @@
-from lib import *
+from lib import get_data
 
-data = open(0).read()
+data = get_data(__file__)
 
-part_2 = True
+t = sum((1 if c == "(" else -1 for c in data))
+print(t)
 
 floor = 0
 for i, c in enumerate(data):
@@ -10,13 +11,6 @@ for i, c in enumerate(data):
         floor += 1
     elif c == ")":
         floor -= 1
-    else:
+    if floor == -1:
-        print(c)
-        assert False
-
-    if part_2 and floor == -1:
         print(i + 1)
         break
-
-if not part_2:
-    print(floor)

2015/d10.py

@@ -1,35 +1,29 @@
-from functools import lru_cache
+from lib import get_data
-data = open(0).read().strip()
 
-part_1 = False
+data = get_data(__file__)
-if part_1:
-    repeats = 40
-else:
-    repeats = 50
 
-@lru_cache
+
-def look_and_say(data: str) -> str:
+def iter(xs):
-    r = ""
+    if not xs:
-    i = 0
+        return []
-    while i < len(data):
+    grouped = []
-        count = 0
+    current_element = xs[0]
-        c = data[i]
+    count = 1
-        while i < len(data) and data[i] == c:
+    for element in xs[1:]:
+        if element == current_element:
             count += 1
-            i += 1
+        else:
-        r += f"{count}{c}"
+            grouped.append(count)
-    return r
+            grouped.append(current_element)
+            current_element = element
+            count = 1
+    grouped.append(count)
+    grouped.append(current_element)
+    return grouped
 
-CHUNK_SIZE = 10000
-for _ in range(repeats):
-    ndata = ""
-    lo, up = 0, CHUNK_SIZE
-    while up < len(data):
-        while up < len(data) and data[up - 1] == data[up]:
-            up += 1
-        ndata += look_and_say(data[lo:up])
-        lo, up = up, up + CHUNK_SIZE
-    ndata += look_and_say(data[lo:up])
-    data = ndata
 
-print(len(data))
+for repeat in [40, 50]:
+    xs = list(map(int, list(data.strip())))
+    for _ in range(repeat):
+        xs = iter(xs)
+    print(len(xs))

2015/d11.py

@@ -1,4 +1,7 @@
-data = open(0).read().strip()
+from lib import get_data
+
+data = get_data(__file__).strip()
+
 
 def is_valid(pw):
     # Passwords must include one increasing straight of at least three letters,
@@ -26,24 +29,26 @@ def is_valid(pw):
         return False
     return True
 
+
 def inc(pw):
     pw = list(map(ord, pw))
     for i in range(len(pw) - 1, -1, -1):
         pw[i] += 1
-        if pw[i] == ord('z') + 1:
+        if pw[i] == ord("z") + 1:
-            pw[i] = ord('a')
+            pw[i] = ord("a")
         else:
             break
     return "".join(map(chr, pw))
 
-part_1 = False
+
+part_1 = True
 valid_count = 0
 while True:
     data = inc(data)
     if is_valid(data):
         valid_count += 1
-    if part_1 and valid_count == 1:
+        if part_1 and valid_count == 1:
-        break
+            print(data)
-    elif valid_count == 2:
+        elif valid_count == 2:
-        break
+            print(data)
-print(data)
+            break

2015/d12.py

@@ -1,27 +1,37 @@
+from lib import get_data
 import json
 
-data = open(0).read().strip()
+data = get_data(__file__).strip()
-part_2 = False
 
-def addup(d):
+o = json.loads(data)
-    r = 0
+
-    if isinstance(d, list):
+
-        for e in d:
+def xsum(obj, part_2=False):
-            v = addup(e)
+    t = 0
-            if v is not None:
+    if type(obj) is int:
-                r += v
+        t += obj
-    elif isinstance(d, dict):
+    elif type(obj) is list:
-        for e in d.values():
+        for o in obj:
-            v = addup(e)
+            t += xsum(o, part_2)
-            if v is None:
+    elif type(obj) is dict:
-                return 0
+        if part_2:
-            r += v
+            st = 0
-    elif isinstance(d, str):
+            for o in obj.values():
-        if part_2 and d == "red":
+                st += xsum(o, part_2)
-            return None
+                if type(o) is str and o == "red":
+                    break
+            else:
+                t += st
+        else:
+            for o in obj.values():
+                t += xsum(o, part_2)
+    elif type(obj) is str:
+        pass
     else:
-        r += d
+        print(obj)
-    return r
+        assert False
+    return t
 
-data = json.loads(data)
+
-print(addup(data))
+print(xsum(o))
+print(xsum(o, True))

2015/d13.py

@@ -1,35 +1,32 @@
+from lib import get_data
+from collections import defaultdict
 from itertools import permutations
-data = open(0).read().strip()
 
-part_2 = False
+data = get_data(__file__).strip()
 
+pairs = defaultdict(int)
 people = set()
-scores = {}
+
 for line in data.splitlines():
-    a, _, wl, score, _, _, _, _, _, _, b = line.split()
+    a, _, gainlose, amount, _, _, _, _, _, _, b = line[:-1].split()
-    b = b.replace(".", "")
+    amount = int(amount)
-    score = int(score)
+    if gainlose == "lose":
-    if wl == "lose":
+        amount = -amount
-        score = -score
+    pairs[(a, b)] = amount
     people.add(a)
-    people.add(b)
-    scores[(a, b)] = score
-    if part_2:
-        scores[(a, "me")] = 0
-        scores[("me", a)] = 0
-        scores[(b, "me")] = 0
-        scores[("me", b)] = 0
 
-if part_2:
-    people.add("me")
 
-max_score = 0
+def calc_gain(people):
-for p in permutations(list(people)):
+    maxgain = 0
-    s = 0
+    for xs in permutations(list(people)):
-    for i in range(len(p)):
+        gain = 0
-        a, b = p[i], p[(i + 1) % len(p)]
+        for i in range(len(xs)):
-        s += scores[(a, b)]
+            gain += pairs[(xs[i], xs[(i + 1) % len(xs)])]
-        s += scores[(b, a)]
+            gain += pairs[(xs[(i + 1) % len(xs)], xs[i])]
-    max_score = max(max_score, s)
+        maxgain = max(maxgain, gain)
+    return maxgain
 
-print(max_score)
+
+print(calc_gain(people))
+people.add("Felix")
+print(calc_gain(people))

2015/d14.py

@@ -1,53 +1,40 @@
-import lib
+from lib import get_data
-data = open(0).read().strip()
+from lib import ints, fst
+from collections import defaultdict
 
-part_1 = False
+data = get_data(__file__).strip()
-
+deers = []
-rds = []
-max_dist = 0
-gtime = 2503
-rds = []
 for line in data.splitlines():
-    speed, time, rest = lib.str_to_ints(line)
+    speed, time, rest = ints(line)
-    rds.append([speed, time, rest, 0, 0, 0])
+    deers.append([0, speed, time, rest, time, 0])
-    speed *= 1000
 
-    dist = 0
+points = defaultdict(int)
-    ctime = 0
+for _ in range(2503):
-    while ctime < gtime:
+    max_dists = defaultdict(list)
-        t = min(time, gtime - ctime)
+    for i, deer in enumerate(deers):
-        dist += (speed * t)
+        dist, speed, time, rest, time_left, rest_left = deer
-        ctime += time
+        if time_left > 0:
-        ctime += rest
+            time_left -= 1
-    max_dist = max(dist, max_dist)
+            dist += speed
+        elif time_left == 0:
+            time_left = -1
+            rest_left = rest - 1
+        elif rest_left > 0:
+            rest_left -= 1
+        elif rest_left == 0:
+            time_left = time - 1
+            dist += speed
+            rest_left = -1
+        max_dists[dist].append(i)
 
-if part_1:
+        deer[0] = dist
-    print(max_dist // 1000)
+        deer[4] = time_left
+        deer[5] = rest_left
 
-scores = [0 for _ in range(len(rds))]
+    max_deers = max(max_dists.items())
-for _ in range(gtime):
+    for di in max_deers[1]:
-    for i in range(len(rds)):
+        points[di] += 1
-        speed, time, rest, dist, travel_time, rest_time = rds[i]
-        rd = rds[i]
-        if travel_time < time:
-            rd[3] += speed
-            rd[4] += 1
-        elif rest_time < rest:
-            rd[5] += 1
-            if rd[5] == rest:
-                rd[4] = 0
-                rd[5] = 0
 
-    max_i = -1
-    max_v = 0
-    for i, rd in enumerate(rds):
-        if rd[3] > max_v:
-            max_i = i
-            max_v = rd[3]
-    for i, rd in enumerate(rds):
-        if rd[3] == max_v:
-            scores[i] += 1
-
-if not part_1:
-    print(max(scores))
 
+print(max(map(fst, deers)))
+print(max(points.values()))

2015/d2.py

@@ -1,19 +1,17 @@
-from lib import *
+from lib import get_data, ints
 
-data = open(0).readlines()
+data = get_data(__file__)
-
-part_1 = True
-if part_1:
-    a = 0
-    for line in data:
-        l, w, h = list(map(int, line.split("x")))
-        slack = min([l * w, w * h, h * l])
-        a +=  (2*l*w + 2*w*h + 2*h*l + slack)
-else: 
-    a = 0
-    for line in data:
-        l, w, h = list(map(int, line.split("x")))
-        sd = min([2 * (l + w), 2 * (w + h), 2 * (h + l)])
-        a +=  sd + (l * w * h)
 
+a = 0
+for line in data.splitlines():
+    l, w, h = ints(line)
+    slack = min([l * w, w * h, h * l])
+    a += 2 * l * w + 2 * w * h + 2 * h * l + slack
+print(a)
+
+a = 0
+for line in data.splitlines():
+    l, w, h = ints(line)
+    sd = min([2 * (l + w), 2 * (w + h), 2 * (h + l)])
+    a += sd + (l * w * h)
 print(a)

2015/d20.py

@@ -4,6 +4,7 @@ import lib
 data = int(open(0).read().strip())
 part_1 = False
 
+
 def calc(n):
     fs = lib.prime_factors(n)
     vs = set([1, n])
@@ -21,6 +22,7 @@ def calc(n):
             r += e * 11
     return r
 
+
 for i in range(3, 10000000):
     c = calc(i)
     if c >= data:

2015/d3.py

@@ -1,9 +1,7 @@
-import sys
+from lib import get_data
-from lib import *
+from lib import V
 
-data = open(0).read()
+data = get_data(__file__)
-
-part_1 = False
 
 DIRS = {
     "^": (-1, 0),
@@ -12,30 +10,27 @@ DIRS = {
     "<": (0, -1),
 }
 
-if part_1:
+pos = V(0, 0)
-    pos = (0, 0)
+poss = set([pos])
-    poss = set([pos])
 
-    for c in data:
+for c in data:
-        d = DIRS[c]
+    d = DIRS[c]
-        pos = pos[0] + d[0], pos[1] + d[1]
+    pos = pos + d
-        poss.add(pos)
+    poss.add(pos)
 
-    print(len(poss))
+print(len(poss))
-    sys.exit(0)
 
-a = (0, 0)
+a = V(0, 0)
-b = (0, 0)
+b = V(0, 0)
 poss = set([a, b])
 for i, c in enumerate(data):
     if i % 2 == 0:
         d = DIRS[c]
-        a = a[0] + d[0], a[1] + d[1]
+        a = a + d
         poss.add(a)
     else:
         d = DIRS[c]
-        b = b[0] + d[0], b[1] + d[1]
+        b = b + d
         poss.add(b)
 
 print(len(poss))
-sys.exit(0)

2015/d4.py

@@ -1,22 +1,15 @@
-from lib import *
+from lib import get_data
 import hashlib
 
-part_1 = True
+data = get_data(__file__).strip()
 
-if part_1:
+for digits in [5, 6]:
-    digits = 5
+    for i in range(10**9):
-else:
+        text = data + str(i)
-    digits = 6
+        md5_hash = hashlib.md5(text.encode()).hexdigest()
-
+        for c in md5_hash[:digits]:
-data = open(0).read().strip()
+            if c != "0":
-
+                break
-for i in range(10**9):
+        else:
-    text = data + str(i)
+            print(i)
-    md5_hash = hashlib.md5(text.encode()).hexdigest()
-    for c in md5_hash[:digits]:
-        if c != "0":
             break
-    else:
-        print(i)
-        break
-

2015/d5.py

@@ -1,53 +1,56 @@
-from lib import *
+from lib import get_data
 
-data = open(0).read()
+data = get_data(__file__)
 
-part_1 = True
-if part_1:
-    res = 0
-    for line in data.splitlines():
-        vc = 0
-        for c in line:
-            if c in "aoeui":
-                vc += 1
-        if vc < 3:
-            continue
 
-        prev = None
+def is_nice(line):
-        for c in line:
+    # It contains at least three vowels (aeiou only), like aei, xazegov, or aeiouaeiouaeiou.
-            if c == prev:
+    vc = 0
-                break
+    for v in "aeiou":
-            prev = c
+        vc += line.count(v)
-        else:
+    if vc < 3:
-            continue
+        return False
 
-        contains = False
+    # It contains at least one letter that appears twice in a row, like xx,
-        ss = ["ab", "cd", "pq", "xy"]
+    # abcdde (dd), or aabbccdd (aa, bb, cc, or dd).
-        for s in ss:
+    for i in range(len(line) - 1):
-            if s in line:
+        if line[i] == line[i + 1]:
-                contains = True
+            break
+    else:
+        return False
 
-        if contains:
+    # It does not contain the strings ab, cd, pq, or xy, even if they are
-            continue
+    # part of one of the other requirements.
-        res += 1
+    for i in range(len(line) - 1):
-    print(res)
+        cc = "".join(line[i : i + 2])
-else:
+        if cc in ["ab", "cd", "pq", "xy"]:
-    res = 0
+            return False
-    for line in data.splitlines():
+    return True
-        pairs = {}
-        for i in range(0, len(line) - 1):
-            p = line[i:i+2]
-            if p in pairs and i > pairs[p] + 1:
-                break
-            if not p in pairs:
-                pairs[p] = i
-        else:
-            continue
 
-        for i in range(0, len(line) - 2):
+
-            if line[i] == line[i + 2]:
+def is_nice_2(line):
-                break
+    good = False
-        else:
+    for i in range(len(line) - 1):
-            continue
+        cc = "".join(line[i : i + 2])
-        res += 1
+        for j in range(i + 2, len(line) - 1):
-    print(res)
+            dd = "".join(line[j : j + 2])
+            if cc == dd:
+                good = True
+    if not good:
+        return False
+
+    for i in range(len(line) - 2):
+        cc = "".join(line[i : i + 3])
+        if cc[0] == cc[2]:
+            break
+    else:
+        return False
+    return True
+
+
+t = sum(1 if is_nice(line) else 0 for line in data.splitlines())
+print(t)
+
+
+t = sum(1 if is_nice_2(line) else 0 for line in data.splitlines())
+print(t)

2015/d6.py

@@ -1,33 +1,29 @@
-from lib import *
+from lib import get_data, ints
+from collections import defaultdict as DD
 
-data = open(0).read()
+data = get_data(__file__)
-
-part_2 = True
-lights = [[0 for _ in range(1000)] for _ in range(1000)]
 
+ons = set()
+ons2 = DD(int)
 for line in data.splitlines():
-    x1, y1, x2, y2 = str_to_ints(line)
+    x1, y1, x2, y2 = ints(line)
     for x in range(x1, x2 + 1):
         for y in range(y1, y2 + 1):
-            if part_2:
+            if "off" in line:
-                if "on" in line:
+                if (x, y) in ons:
-                    lights[x][y] += 1
+                    ons.remove((x, y))
-                elif "off" in line:
+                ons2[(x, y)] = max(ons2[(x, y)] - 1, 0)
-                    lights[x][y] = max(0, lights[x][y] - 1)
+            elif "on" in line:
+                ons.add((x, y))
+                ons2[(x, y)] += 1
+            elif "toggle" in line:
+                if (x, y) in ons:
+                    ons.remove((x, y))
                 else:
-                    lights[x][y] += 2
+                    ons.add((x, y))
+                ons2[(x, y)] += 2
             else:
-                if "on" in line:
+                assert False
-                    lights[x][y] = 1
-                elif "off" in line:
-                    lights[x][y] = 0
-                else:
-                    if lights[x][y] == 1:
-                        lights[x][y] = 0
-                    else:
-                        lights[x][y] = 1
 
-res = 0
+print(len(ons))
-for row in lights:
+print(sum(ons2.values()))
-    res += sum(row)
-print(res)

2015/d7.py

@@ -1,90 +1,54 @@
-from lib import *
+from lib import get_data
 
-data = open(0).read()
+data = get_data(__file__)
 
-part_2 = False
-if part_2:
-    gates = {"b": 16076}
-else:
-    gates = {}
 
-while "a" not in gates:
+def run(wires={}):
-    for line in data.splitlines():
+    def get(a):
-        lhs, rhs = line.split(" -> ")
+        try:
-        if part_2 and rhs == "b":
+            return int(a)
-            continue
+        except ValueError:
-        lhs = lhs.strip()
+            pass
-        if "NOT" in lhs:
+        if a in wires:
-            op, op1 = lhs.split(" ")
+            return wires[a]
-            assert op == "NOT"
-            try:
-                op1 = int(op1)
-                gates[rhs] = ~op1 & 0xffff
-            except ValueError:
-                if op1 in gates and isinstance(gates[op1], int):
-                    gates[rhs] = ~gates[op1] & 0xffff
-        elif "OR" in lhs:
-            op1, op, op2 = lhs.split(" ")
-            assert op == "OR"
-            try:
-                op1 = int(op1)
-            except ValueError:
-                if op1 in gates and isinstance(gates[op1], int):
-                    op1 = gates[op1]
-            try:
-                op2 = int(op2)
-            except ValueError:
-                if op2 in gates and isinstance(gates[op2], int):
-                    op2 = gates[op2]
-
-            if not (isinstance(op1, int) and isinstance(op2, int)):
-                continue
-            gates[rhs] = (op1 | op2) 
-        elif "AND" in lhs:
-            op1, op, op2 = lhs.split(" ")
-            assert op == "AND"
-            try:
-                op1 = int(op1)
-            except ValueError:
-                if op1 in gates and isinstance(gates[op1], int):
-                    op1 = gates[op1]
-            try:
-                op2 = int(op2)
-            except ValueError:
-                if op2 in gates and isinstance(gates[op2], int):
-                    op2 = gates[op2]
-
-            if not (isinstance(op1, int) and isinstance(op2, int)):
-                continue
-            gates[rhs] = (op1 & op2) 
-        elif "LSHIFT" in lhs:
-            op1, op, op2 = lhs.split(" ")
-            op2 = int(op2)
-            try:
-                op1 = int(op1)
-            except ValueError:
-                if op1 in gates:
-                    op1 = gates[op1]
-                else:
-                    continue
-            gates[rhs] = (op1 << op2) & 0xffff
-        elif "RSHIFT" in lhs:
-            op1, op, op2 = lhs.split(" ")
-            op2 = int(op2)
-            try:
-                op1 = int(op1)
-            except ValueError:
-                if op1 in gates:
-                    op1 = gates[op1]
-                else:
-                    continue
-            gates[rhs] = (op1 >> op2) & 0xffff
         else:
-            try:
+            return None
-                lhs = int(lhs)
-                gates[rhs] = lhs
-            except ValueError:
-                if lhs in gates:
-                    gates[rhs] = gates[lhs]
 
-print(gates["a"])
+    while "a" not in wires:
+        for line in data.splitlines():
+            lhs, rhs = line.split(" -> ")
+            if rhs in wires:
+                continue
+            match lhs.split():
+                case [a, "AND", b]:
+                    a, b = get(a), get(b)
+                    if a is not None and b is not None:
+                        wires[rhs] = a & b
+                case [a, "OR", b]:
+                    a, b = get(a), get(b)
+                    if a is not None and b is not None:
+                        wires[rhs] = a | b
+                case [a, "LSHIFT", b]:
+                    a, b = get(a), get(b)
+                    if a is not None and b is not None:
+                        wires[rhs] = a << b
+                case [a, "RSHIFT", b]:
+                    a, b = get(a), get(b)
+                    if a is not None and b is not None:
+                        wires[rhs] = a >> b
+                case ["NOT", a]:
+                    a = get(a)
+                    if a is not None:
+                        wires[rhs] = ~a & 0xFFFF
+                case [a]:
+                    a = get(a)
+                    if a is not None:
+                        wires[rhs] = a
+    return wires
+
+
+a = run()["a"]
+print(a)
+
+a2 = run(wires={"b": a})["a"]
+print(a2)

2015/d8.py

@@ -1,26 +1,25 @@
-from lib import *
+from lib import get_data
+import re
 
-data = open(0).read()
+data = get_data(__file__)
-part_1 = False
 
-if part_1:
+r1 = re.compile(r"\\x[0-9a-f][0-9a-f]")
-    r1 = re.compile(r"\\x[0-9a-f][0-9a-f]")
+r2 = re.compile(r"\\\\")
-    r2 = re.compile(r"\\\\")
+r3 = re.compile(r"\\\"")
-    r3 = re.compile(r"\\\"")
 
-    enc, mem = 0, 0
+enc, mem = 0, 0
-    for line in data.splitlines():
+for line in data.splitlines():
-        mem += len(line)
+    mem += len(line)
-        line = r1.sub("^", line)
+    line = r1.sub("^", line)
-        line = r2.sub("^", line)
+    line = r2.sub("^", line)
-        line = r3.sub("^", line)
+    line = r3.sub("^", line)
-        enc += len(line) - 2
+    enc += len(line) - 2
-    print(mem - enc)
+print(mem - enc)
-else:
+
-    ori, enc = 0, 0
+ori, enc = 0, 0
-    for line in data.splitlines():
+for line in data.splitlines():
-        ori += len(line)
+    ori += len(line)
-        line = line.replace("\\", "\\\\")
+    line = line.replace("\\", "\\\\")
-        line = line.replace("\"", "\\\"")
+    line = line.replace('"', '\\"')
-        enc += len(line) + 2
+    enc += len(line) + 2
-    print(enc - ori)
+print(enc - ori)

2015/d9.py

@@ -1,42 +1,27 @@
-from lib import *
+from lib import get_data
-from typing import Iterator
+from itertools import permutations
 
-# Just for fun instead of using itertools.permutations.
+data = get_data(__file__)
-def permutations(xs) -> Iterator:
-    assert len(xs) > 0
-    if len(xs) == 1:
-        yield xs
-    else:
-        x = xs.pop()
-        for p in permutations(xs):
-            for i in range(len(p) + 1):
-                pn = list(p)
-                pn.insert(i, x)
-                yield pn
-
-part_1 = False
-data = open(0).read()
 
 dists = {}
 nodes = set()
 for line in data.splitlines():
-    path, dist = line.split(" = ")
+    a, _, b, _, d = line.split()
-    dist = int(dist)
-    a, b = path.split(" to ")
     nodes.add(a)
     nodes.add(b)
+    dist = int(d)
     dists[(a, b)] = dist
     dists[(b, a)] = dist
 
-if part_1:
+d_min = 10**12
-    mdist = 10**12
+d_max = 0
-    for route in permutations(nodes):
+for p in permutations(list(nodes)):
-        dist = sum([dists[(route[i], route[i + 1])] for i in range(len(route) - 1)])
+    d = 0
-        mdist = min(dist, mdist)
+    for i in range(len(p) - 1):
-    print(mdist)
+        a, b = p[i], p[i + 1]
-else:
+        d += dists[(a, b)]
-    mdist = 0
+    d_min = min(d, d_min)
-    for route in permutations(nodes):
+    d_max = max(d, d_max)
-        dist = sum([dists[(route[i], route[i + 1])] for i in range(len(route) - 1)])
+
-        mdist = max(dist, mdist)
+print(d_min)
-    print(mdist)
+print(d_max)

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})"