Solve problem 381

python/e381.py

@@ -0,0 +1,42 @@
+from lib_prime import primes
+from lib_misc import modinv
+
+
+def s(p: int) -> int:
+    a = p - 1
+    fp = p - 1
+    r = fp
+    # Calculate (!(p - 1) + !(p - 2) + ... + !(p - 5)) % p
+    for _ in range(4):
+        fp = fp * modinv(a, p) % p
+        a -= 1
+        r += fp
+        r %= p
+    return r
+
+
+def euler_381():
+    assert s(5) == 4
+    assert s(7) == 4
+
+    # Example given by problem statement
+    t = 0
+    for p in primes(100):
+        if p < 5:
+            continue
+        t += s(p)
+    assert t == 480
+
+    # Actual solution (#slow)
+    t = 0
+    for p in primes(10**8):
+        if p < 5:
+            continue
+        t += s(p)
+    return t
+
+
+if __name__ == "__main__":
+    solution = euler_381()
+    print(f"e381.py: {solution}")
+    assert solution == 139602943319822

python/e961.py

@@ -0,0 +1,50 @@
+from functools import cache
+
+
+@cache
+def remove(n: str, pos: int) -> str:
+    assert pos < len(n)
+    r = n[:pos] + n[pos + 1:]
+    r = r.lstrip("0")
+    return r
+
+
+@cache
+def player_wins(n: str) -> bool:
+    if len(n) == 1:
+        return True
+    for i in range(len(n)):
+        if not player_wins(remove(n, i)):
+            return True
+    return False
+
+def w(n: int) -> int:
+    r = 0
+    xs = [("", 0)]
+    for _ in range(n):
+        nxs = []
+        for x, x_count in xs:
+            nxs.append(("0" + x, x_count))
+            nx, nx_count = "x" + x, x_count + 1
+            if player_wins(nx):
+                r += (9 ** nx_count)
+            nxs.append((nx, nx_count))
+        xs = nxs
+    return r
+
+
+def euler_961():
+    assert remove("105", 0) == "5"
+    assert remove("105", 1) == "15"
+    assert remove("105", 2) == "10"
+    assert remove("10005", 0) == "5"
+    assert w(2) == 18
+    assert w(4) == 1656
+    return w(18)
+
+
+if __name__ == "__main__":
+    solution = euler_961()
+    print("e961.py: " + str(solution))
+    # assert(solution == 0)
+

python/lib_create_templates.py

@@ -7,7 +7,7 @@ def euler_XXX():
 
 if __name__ == "__main__":
     solution = euler_XXX()
-    print("eXXX.py: " + str(solution))
+    print(f"eXXX.py: {solution}")
     # assert(solution == 0)
 
 """

python/lib_misc.py

@@ -230,3 +230,18 @@ def cache(f):
         return r
     return func_cached
 
+
+def ext_gcd(a: int, b: int) -> tuple[int, int, int]:
+    if a == 0:
+        return (b, 0, 1)
+    else:
+        gcd, x, y = ext_gcd(b % a, a)
+        return (gcd, y - (b // a) * x, x)
+
+
+def modinv(a: int, b: int) -> int:
+    gcd, x, _ = ext_gcd(a, b)
+    if gcd != 1:
+        raise Exception('Modular inverse does not exist')
+    else:
+        return x % b