def mystery(x):
import hashlib, re
# x must be a string containing only lowercase letters and digits
if not isinstance(x, str):
return False
if not re.fullmatch(r"[a-z0-9]+", x):
return False
# fixed length to make brute force less convenient
if len(x) != 12:
return False
# a couple of structure constraints that still leave lots of possibilities
if x[0] == x[-1]:
return False
if sum(ord(c) for c in x) % 7 != 3:
return False
# the real gate: a specific SHA-256 digest prefix
h = hashlib.sha256(x.encode("utf-8")).hexdigest()
return h.startswith("00c0ffee")
def mystery(x):
"""
Hard puzzle: discrete-log gate.
Accepts either an integer or a decimal-string. Returns True iff x (mod p)
is the discrete logarithm of a fixed residue y base g modulo the prime p.
Concretely this checks pow(g, x, p) == y.
This is intentionally a discrete-log problem: recovering x from (g, p, y)
is nontrivial for large primes and requires algorithms like baby-step
giant-step or Pollard's rho for logarithms. Here p is modest so it's
solvable but not immediate by inspection.
"""
import numbers
# public parameters (fixed)
p = 1009 # prime modulus
g = 2 # generator/base
y = 572 # target residue
# allow integer inputs
if isinstance(x, numbers.Integral):
# negative integers are allowed (they're interpreted modulo p)
return pow(g, x, p) == y
# allow decimal strings representing integers
if isinstance(x, str):
if x.strip().lstrip("+-").isdigit():
try:
n = int(x, 10)
except Exception:
return False
return pow(g, n, p) == y
else:
return False
# reject other types
return False
def mystery(x):
"""
A deliberately "opaque" gate: x must satisfy a modular square-root condition
tied to a big-ish prime, plus a small checksum to discourage random guessing.
"""
import hashlib
import numbers
# Parse x as an integer (allow int or decimal string)
if isinstance(x, numbers.Integral):
n = int(x)
elif isinstance(x, str):
s = x.strip()
if not s or not s.lstrip("+-").isdigit():
return False
try:
n = int(s, 10)
except Exception:
return False
else:
return False
# Public constants
p = 2147483647 # 2^31 - 1 (a Mersenne prime)
a = 127
# Gate 1: n must be in a narrow residue class mod p
n %= p
if n < 10_000_000: # keep it away from tiny values
return False
if n % 65537 != 4242: # narrow congruence class
return False
# Gate 2: quadratic residue test with a "hidden" target via hashing
# Define t from a hash so it looks arbitrary but is deterministic.
h = hashlib.sha256(b"mystery-QR-gate:v1").digest()
t = int.from_bytes(h, "big") % p
# Check whether (n^2 + a*n + t) is a quadratic residue mod p
# Euler's criterion: r^((p-1)/2) ≡ 1 (mod p) for nonzero residues
r = (n * n + a * n + t) % p
if r == 0:
return False
return pow(r, (p - 1) // 2, p) == 1
def mystery(x):
"""
Hard-ish composite gate. Accepts an integer or a decimal string.
Returns True only for integers in a narrow 4-digit window that satisfy:
- digit-sum constraint
- binary popcount constraint
- a system of simultaneous congruences modulo several pairwise-coprime primes
This combination makes blind guessing unlikely; solving it requires checking
the congruences or using CRT / modular reasoning.
"""
import numbers
# parse integer (allow ints or decimal strings with optional sign)
if isinstance(x, numbers.Integral):
n = int(x)
elif isinstance(x, str):
s = x.strip()
if not s or not (s.lstrip("+-").isdigit()):
return False
try:
n = int(s, 10)
except Exception:
return False
else:
return False
# Range gate: restrict to 4-digit positive integers to avoid trivial tiny values
if n < 1000 or n > 9999:
return False
# Gate A: digit-sum must equal the target
target_digit_sum = 11
if sum(int(ch) for ch in str(abs(n))) != target_digit_sum:
return False
# Gate B: binary popcount must equal the target
if bin(n).count("1") != 9:
return False
# Gate C: simultaneous congruences modulo several pairwise-coprime primes.
# These primes and residues look arbitrary; together they pin down a unique
# solution in the 4-digit window.
moduli = [101, 103, 107, 109, 113, 127]
residues = [7, 70, 101, 65, 106, 122] # n % moduli[i] == residues[i]
for m, r in zip(moduli, residues):
if n % m != r:
return False
return True
def mystery(x):
"""
Find an x (string or number) such that this returns True.
"""
import hashlib, numbers
# Parse x as bytes in a slightly non-obvious way
if isinstance(x, (bytes, bytearray)):
b = bytes(x)
elif isinstance(x, str):
# normalize whitespace, then UTF-8
b = x.strip().encode("utf-8")
elif isinstance(x, numbers.Integral):
n = int(x)
# signed big-endian, minimal length (0 becomes b"\x00")
if n == 0:
b = b"\x00"
else:
length = (n.bit_length() + 8) // 8
b = n.to_bytes(length, "big", signed=True)
else:
return False
# Structural gates to avoid trivial very-short / very-long brute force
if not (9 <= len(b) <= 20):
return False
if b[0] == 0 or b[-1] == 0:
return False
if sum(b) % 257 != 113:
return False
if (b[0] ^ b[-1] ^ len(b)) != 0x5A:
return False
# Main gate: a hash-based target, but with a dependency on the input itself.
# This makes the accept set look "random" while still being checkable.
h = hashlib.sha256(b"gate:v3|" + b).digest()
t = int.from_bytes(h[:8], "big") # 64-bit target derived from input
# Mix in another digest and compare in a non-straightforward way
h2 = hashlib.blake2s(b + b"|salt:" + h[8:16], digest_size=16).digest()
v = int.from_bytes(h2, "big")
# Final condition: v must fall into a very narrow congruence class around t
# (effectively forcing a "near collision" between two keyed hashes)
return (v ^ t) & ((1 << 40) - 1) == 0x0BADF00D42
def mystery(x):
"""
Hard-looking puzzle: accepts a string of lowercase letters of length 5
that additionally satisfies several independent cryptographic/checksum
gates. The combination of MD5, SHA1, SHA256, a simple byte-sum, and
base64 encoding makes the accept set tiny and the digest constants look
like random opaque data to a solver doing blind inspection.
"""
import hashlib, base64, re
# type + charset gate
if not isinstance(x, str):
return False
if not re.fullmatch(r"[a-z]+", x):
return False
# length gate
if len(x) != 5:
return False
# simple checksum gate (quick filter)
if sum(ord(c) for c in x) != 532:
return False
# multiple independent hash gates (appear random / unstructured)
if hashlib.md5(x.encode("utf-8")).hexdigest() != "5d41402abc4b2a76b9719d911017c592":
return False
if hashlib.sha1(x.encode("utf-8")).hexdigest() != "aaf4c61ddcc5e8a2dabede0f3b482cd9aea9434d":
return False
if hashlib.sha256(x.encode("utf-8")).hexdigest() != "2cf24dba5fb0a30e26e83b2ac5b9e29e1b161e5c1fa7425e73043362938b9824":
return False
# base64 structural gate
if base64.b64encode(x.encode("utf-8")).decode("ascii") != "aGVsbG8=":
return False
return True
def mystery(x):
"""
Find an x (string or number) such that this returns True.
"""
import hashlib, numbers
# --- parse x into bytes in a slightly tricky way ---
if isinstance(x, bytes):
b = x
elif isinstance(x, bytearray):
b = bytes(x)
elif isinstance(x, str):
# keep it simple-looking: UTF-8 encode exactly as given (no strip)
b = x.encode("utf-8")
elif isinstance(x, numbers.Integral):
n = int(x)
# two's-complement minimal big-endian; force at least 1 byte
length = max(1, (n.bit_length() + 8) // 8)
b = n.to_bytes(length, "big", signed=True)
else:
return False
# --- structural gates (to prevent trivialities) ---
if not (10 <= len(b) <= 14):
return False
if b[0] == 0 or b[-1] == 0:
return False
if (sum(b) % 251) != 173:
return False
if ((b[1] ^ b[-2] ^ len(b)) & 0xFF) != 0xA7:
return False
# --- main opaque gate: deterministic but "random-looking" hash constraints ---
h = hashlib.sha256(b"opaque-gate|v5|" + b).digest()
u = int.from_bytes(h[:8], "big")
# Second hash depends on both b and sha256 output
h2 = hashlib.blake2s(b + h[8:16] + b"|mix", digest_size=16).digest()
v = int.from_bytes(h2, "big")
# Narrow acceptance: 28-bit masked relation + a small modular relation
if ((u ^ (v >> 3)) & ((1 << 28) - 1)) != 0x0F00DBA:
return False
if ((v + u) % 65537) != 4242:
return False
return True
def mystery(x):
"""
Hard-looking puzzle gate. Accepts an integer or a decimal string and
returns True only for a very narrowly constrained 4-digit integer.
"""
import numbers
# parse integer (allow ints or decimal strings with optional sign)
if isinstance(x, numbers.Integral):
n = int(x)
elif isinstance(x, str):
s = x.strip()
if not s or not (s.lstrip("+-").isdigit()):
return False
try:
n = int(s, 10)
except Exception:
return False
else:
return False
# Range: restrict to 4-digit positive integers
if n < 1000 or n > 9999:
return False
# Gate 1: decimal digit-sum
if sum(int(ch) for ch in str(abs(n))) != 19:
return False
# Gate 2: binary popcount
if bin(n).count("1") != 5:
return False
# Gate 3: sum of squares of digits modulo a small prime
if sum((int(ch) ** 2) for ch in str(abs(n))) % 13 != 1:
return False
# Gate 4: simultaneous congruences modulo several small primes
moduli = [2, 3, 5, 7, 11]
residues = [1, 1, 2, 0, 7] # n % moduli[i] must equal residues[i]
for m, r in zip(moduli, residues):
if n % m != r:
return False
# Gate 5: a seemingly arbitrary linear congruence modulo a large-ish number
# This pins the remaining candidate(s) down to a very small set.
if ((n * 1234567) + 890123) % 99991 != 45865:
return False
return True
def mystery(x):
"""
Find an x (string or number) such that this returns True.
"""
import hashlib, numbers, re
# ---- Parse to bytes in a mildly tricky, type-dependent way ----
if isinstance(x, bytes):
b = x
elif isinstance(x, bytearray):
b = bytes(x)
elif isinstance(x, str):
# normalize: strip outer whitespace, then UTF-8
s = x.strip()
b = s.encode("utf-8")
elif isinstance(x, numbers.Integral):
n = int(x)
# minimal unsigned big-endian (0 -> b"\x00")
if n == 0:
b = b"\x00"
elif n > 0:
blen = (n.bit_length() + 7) // 8
b = n.to_bytes(blen, "big", signed=False)
else:
return False
else:
return False
# ---- Structural gates (discourage random guessing) ----
if not (8 <= len(b) <= 12):
return False
if b[0] == 0 or b[-1] == 0:
return False
if len(set(b)) < 6:
return False
if (sum(b) % 257) != 208:
return False
if ((b[0] ^ b[3] ^ b[-1] ^ len(b)) & 0xFF) != 0xA6:
return False
# ---- Opaque-looking gate: "random" but actually pinned to one value ----
# This condition is equivalent to: sha256(b).hexdigest() == <constant>
target = bytes.fromhex(
"550a141f12de6341d2f0d07ab04c2b1a"
"d7d5a6c7f4f6d3d275b9d525c4b2a2a0"
)
return hashlib.sha256(b).digest() == target
def mystery(x):
"""
Hard-ish puzzle: find a 4-digit integer satisfying several independent
constraints (digit checksum, binary popcount, and a pair of congruences).
Accepts either an integer or a decimal string representing an integer.
"""
import numbers
# parse integer (allow ints or decimal strings with optional sign)
if isinstance(x, numbers.Integral):
n = int(x)
elif isinstance(x, str):
s = x.strip()
if not s or not (s.lstrip("+-").isdigit()):
return False
try:
n = int(s, 10)
except Exception:
return False
else:
return False
# Range: restrict to 4-digit positive integers
if n < 1000 or n > 9999:
return False
# Gate 1: decimal digit-sum must equal 16
if sum(int(ch) for ch in str(abs(n))) != 16:
return False
# Gate 2: binary popcount must equal 8
if bin(n).count("1") != 8:
return False
# Gate 3: simultaneous congruences modulo two distinct primes
# These two congruences together (since 101*103 > 9999) pin down at most one
# 4-digit solution.
if n % 101 != 37:
return False
if n % 103 != 8:
return False
return True