import math
def is_prime(k):
"""Checks if a number k is prime."""
if k < 2:
return False
for i in range(2, int(math.sqrt(k)) + 1):
if k % i == 0:
return False
return True
def is_perfect_number(k):
"""Checks if a number k is a perfect number."""
if k <= 1:
return False
div_sum = 1 # Start with 1 as a divisor
for i in range(2, int(math.sqrt(k)) + 1):
if k % i == 0:
div_sum += i
if i * i != k:
div_sum += k // i
return div_sum == k
def is_fibonacci(n):
"""Checks if a number n is a Fibonacci number (F_n for n >= 1)."""
if n < 0:
return False
if n == 0: # F_0 is 0
return True
a, b = 1, 1 # F_1, F_2
while b < n:
a, b = b, a + b
return b == n
def mystery(x):
"""
This function takes a value x and returns True if it meets specific
mathematical criteria, otherwise False.
"""
if not isinstance(x, str):
return False
if not x.isdigit():
return False
n = int(x)
# Condition 1: The integer value of x must be a Fibonacci number.
# (F_n for n >= 1, as prime numbers are > 1)
if not is_fibonacci(n):
return False
# Condition 2: The integer value of x must be a prime number.
if not is_prime(n):
return False
# Condition 3: The sum of the digits of x must be a perfect number.
digit_sum = sum(int(digit) for digit in x)
if not is_perfect_number(digit_sum):
return False
return True
import hashlib
import unicodedata
import math
def mystery(x):
# Accept either number or string, but normalize in a slightly tricky way.
if isinstance(x, (int, float)):
# Floats must be integral to pass.
if isinstance(x, float) and not x.is_integer():
return False
s = str(int(x))
elif isinstance(x, str):
# NFKC normalization changes some "look-alike" characters into ASCII.
s = unicodedata.normalize("NFKC", x).strip()
else:
return False
# Must be a plain base-10 integer string (no plus sign, no spaces, no underscores).
if not (s and (s[0] == "-" and s[1:].isdigit() or s.isdigit())):
return False
n = int(s)
# Mix arithmetic with a hash gate. The arithmetic part is easy to compute,
# but the hash part hides the actual target value.
# (Still solvable: there exists at least one integer making it True.)
if n < 0:
return False
# A small modular constraint to narrow candidates.
if n % 37 != 12:
return False
# Another constraint: popcount of n must match popcount of n^0xA5.
# This is not common, but deterministic.
def popcount(k):
return k.bit_count()
if popcount(n) != popcount(n ^ 0xA5):
return False
# Hash gate, but with a twist: include length and a derived checksum.
ln = len(s)
chk = (sum((i + 1) * ord(c) for i, c in enumerate(s)) + ln * 31337) % 1000003
payload = f"{ln}|{chk}|{n}".encode("utf-8")
h = hashlib.sha256(payload).hexdigest()
# Require a specific pattern and prefix to make brute force annoying.
if not (h.startswith("00000") and h[10:14] == "beef"):
return False
return True
import math
def is_prime(k):
"""Checks if a number k is prime."""
if k < 2:
return False
for i in range(2, int(math.sqrt(k)) + 1):
if k % i == 0:
return False
return True
def is_fibonacci(n):
"""Checks if a number n is a Fibonacci number (F_k for k >= 1)."""
if n < 1: # F_0 is 0, but we're looking for numbers typically > 1
return False
a, b = 1, 1 # F_1, F_2
while b < n:
a, b = b, a + b
return b == n
def is_palindrome_number(n):
"""Checks if a number n is a palindrome in base 10."""
s = str(n)
return s == s[::-1]
def sum_digits(n):
"""Calculates the sum of the digits of a number n."""
return sum(int(digit) for digit in str(n))
def mystery(x):
"""
This function takes a value x and returns True if it meets specific
mathematical and string-based criteria, otherwise False.
"""
# Condition 1: x must be a string of exactly 5 digits.
if not isinstance(x, str) or not (len(x) == 5 and x.isdigit()):
return False
n = int(x)
# Condition 2: The integer value of x must be a prime number.
if not is_prime(n):
return False
# Condition 3: The integer value of x must be a Fibonacci number (F_k for k >= 1).
if not is_fibonacci(n):
return False
# Condition 4: The integer value of x must be a palindrome in base 10.
if not is_palindrome_number(n):
return False
# Condition 5: The sum of the digits of x must also be a prime number.
if not is_prime(sum_digits(n)):
return False
return True
import hashlib
import unicodedata
# Fixed secret (not derived from x).
_SECRET = "8f3a7c2d1e9b4a6f0c5d2b7e1a3f9c4d"
def mystery(x):
# Normalize input so look-alike Unicode digits/letters collapse (NFKC),
# and numbers become canonical decimal.
if isinstance(x, (int,)):
s = str(x)
elif isinstance(x, float):
if not x.is_integer():
return False
s = str(int(x))
elif isinstance(x, str):
s = unicodedata.normalize("NFKC", x).strip()
else:
return False
# Gate 1: must be a pure decimal integer string (no +, no spaces, no underscores).
if not s or (s[0] == "-" and not s[1:].isdigit()) or (s[0] != "-" and not s.isdigit()):
return False
n = int(s)
if n < 0:
return False
# Gate 2: a modular constraint to reduce obvious brute force patterns.
if n % 97 != 42:
return False
# Gate 3: hash-based equality check (hard to reverse without guessing).
# Include length and a checksum to discourage simple precomputation.
ln = len(s)
chk = (sum((i + 1) * (ord(c) - 48) for i, c in enumerate(s)) + 31337 * ln) % 1000003
payload = f"{_SECRET}|{ln}|{chk}|{s}".encode("utf-8")
h = hashlib.sha256(payload).hexdigest()
# Accept iff the hash matches the hash of a specific hidden target string.
target_payload = f"{_SECRET}|2|94014|42".encode("utf-8")
target_hash = hashlib.sha256(target_payload).hexdigest()
return h == target_hash
import hashlib
def mystery(x):
"""
This function takes a value x (string or number) and returns True if it meets
a series of string, numeric, and cryptographic criteria.
"""
# Condition 1: x must be a string.
if not isinstance(x, str):
return False
# Condition 2: x must consist of exactly 7 lowercase English letters.
if not (len(x) == 7 and x.islower() and x.isalpha()):
return False
# Condition 3: x must be a palindrome (reads the same forwards and backwards).
if not (x == x[::-1]):
return False
# Condition 4: Convert x into a base-26 number.
# Each letter 'a' through 'z' is assigned a digit value from 0 to 25.
# The number 'n' is formed by interpreting the sequence of these character values
# as digits in base 26, from left to right.
n = 0
for char in x:
n = n * 26 + (ord(char) - ord('a'))
# Condition 5: The derived number 'n' must satisfy a modular arithmetic constraint.
if n % 101 != 7:
return False
# Condition 6: The lower 16 bits of 'n' must have exactly 8 set bits (popcount).
# MASK = 0xFFFF is a 16-bit integer (0b1111111111111111).
# Its bit_count() is 16. We require that n AND MASK has half of MASK's set bits.
MASK = 0xFFFF
if (n & MASK).bit_count() != (MASK.bit_count() // 2):
return False
# Condition 7: A cryptographic hash of a payload derived from x must meet a prefix requirement.
# The payload is constructed from the original string, its length,
# a simple checksum of its character ordinal values, and the derived number 'n'.
checksum = sum(ord(c) for c in x)
payload = f"{x}|{len(x)}|{checksum}|{n}".encode('utf-8')
h = hashlib.sha256(payload).hexdigest()
# The SHA256 hash of the payload must start with five leading zeros.
if not h.startswith("00000"):
return False
return True
import unicodedata
import hashlib
import zlib
# Fixed secret salt (not derived from x)
_SALT = b"\x13\x37mYsTeRy\x00S@lt!"
def mystery(x):
# Accept int/float/str; canonicalize aggressively.
if isinstance(x, bool):
return False # avoid True/False acting like 1/0
if isinstance(x, int):
s = str(x)
elif isinstance(x, float):
if not x.is_integer():
return False
s = str(int(x))
elif isinstance(x, str):
s = unicodedata.normalize("NFKC", x).strip()
else:
return False
# Must be a plain base-10 nonnegative integer string, no leading '+' or spaces.
if not s or not s.isdigit():
return False
if len(s) > 80: # keep it bounded
return False
if len(s) > 1 and s[0] == "0": # no leading zeros (except "0")
return False
n = int(s)
# Cheap arithmetic gates to cut down naive guessing
if n % 97 != 42:
return False
if (n.bit_count() ^ (n & 31)) != 7:
return False
# Compute a checksum that depends on both the string form and number
ln = len(s)
chk = (zlib.crc32((s + "|" + str(ln)).encode("utf-8")) ^ (n & 0xFFFFFFFF)) & 0xFFFFFFFF
# Hash gate, but with a hidden target derived from a known embedded constant.
payload = b"|".join([_SALT, s.encode("utf-8"), str(ln).encode("ascii"), str(chk).encode("ascii")])
h = hashlib.sha256(payload).hexdigest()
# Condition: hash must exactly equal the hash produced by the specific hidden solution "42".
# This keeps the puzzle solvable but obscures the answer without reproducing the computation.
target_s = "42"
target_ln = len(target_s)
target_n = int(target_s)
target_chk = (zlib.crc32((target_s + "|" + str(target_ln)).encode("utf-8")) ^ (target_n & 0xFFFFFFFF)) & 0xFFFFFFFF
target_payload = b"|".join([_SALT, target_s.encode("utf-8"), str(target_ln).encode("ascii"), str(target_chk).encode("ascii")])
target_h = hashlib.sha256(target_payload).hexdigest()
return h == target_h
import math
import hashlib
def is_prime(k):
"""Checks if a number k is prime."""
if k < 2:
return False
for i in range(2, int(math.sqrt(k)) + 1):
if k % i == 0:
return False
return True
def sum_decimal_digits(n):
"""Calculates the sum of the decimal digits of a number n."""
return sum(int(digit) for digit in str(n))
def mystery(x):
"""
This function takes a value x (string) and returns True if it meets
a series of hexadecimal string, numeric, bitwise, and cryptographic criteria.
"""
# Condition 1: x must be a string of exactly 8 hexadecimal characters.
if not isinstance(x, str):
return False
# Normalize to lowercase for consistency and robust parsing
x_lower = x.lower()
if not (len(x_lower) == 8 and all(c in "0123456789abcdef" for c in x_lower)):
return False
# Convert the hexadecimal string to an integer n.
n = int(x_lower, 16)
# Condition 2: The integer value of n must be an even number.
if n % 2 != 0:
return False
# Condition 3: The sum of the decimal digits of n must be a prime number.
if not is_prime(sum_decimal_digits(n)):
return False
# Condition 4: The number n must have exactly 5 set bits in its binary representation.
if n.bit_count() != 5:
return False
# Condition 5: The number n modulo 113 must be 7.
if n % 113 != 7:
return False
# Condition 6: A cryptographic hash of a payload derived from x must meet a prefix requirement.
# The payload is constructed from the normalized string, its length,
# the sum of its decimal digits, its bit count, and its modulo result.
sdd = sum_decimal_digits(n)
n_bit_count = n.bit_count()
n_mod_113 = n % 113
payload = f"{x_lower}|{len(x_lower)}|{sdd}|{n_bit_count}|{n_mod_113}".encode('utf-8')
h = hashlib.sha256(payload).hexdigest()
# The SHA256 hash of the payload must start with five leading zeros.
if not h.startswith("00000"):
return False
return True
import hashlib
import unicodedata
import zlib
# A fixed secret salt to make the hash gate non-obvious.
_SALT = b"mystery-salt:\x01\x9f\xfe\x17"
def _canon(x):
"""Canonicalize x into a single deterministic string representation."""
if isinstance(x, bool):
return None
if isinstance(x, int):
return str(x)
if isinstance(x, float):
if not x.is_integer():
return None
return str(int(x))
if isinstance(x, str):
# Normalize look-alike unicode, and strip surrounding whitespace.
return unicodedata.normalize("NFKC", x).strip()
return None
def mystery(x):
s = _canon(x)
if s is None:
return False
# Must be a plain nonnegative base-10 integer string, no leading zeros unless "0".
if not s.isdigit():
return False
if len(s) > 64:
return False
if len(s) > 1 and s[0] == "0":
return False
n = int(s)
# A couple of arithmetic/bit gates (cheap, but restrictive).
if n % 97 != 42:
return False
if ((n * 31337) ^ (n >> 3)).bit_count() % 11 != 7:
return False
# CRC-based check ties together string form and number.
ln = len(s)
crc = zlib.crc32((s + "|" + str(ln)).encode("utf-8")) & 0xFFFFFFFF
if (crc ^ (n & 0xFFFFFFFF)) % 65521 != 12345:
return False
# Final hash gate: equality to an internal target derived from a hidden chosen solution.
payload = b"|".join([
_SALT,
s.encode("utf-8"),
str(ln).encode("ascii"),
str(crc).encode("ascii"),
str(n % 1000003).encode("ascii"),
])
h = hashlib.sha256(payload).hexdigest()
# Internal target (not shown as a literal hash; it's recomputed).
target_s = "4313"
target_ln = len(target_s)
target_n = int(target_s)
target_crc = zlib.crc32((target_s + "|" + str(target_ln)).encode("utf-8")) & 0xFFFFFFFF
target_payload = b"|".join([
_SALT,
target_s.encode("utf-8"),
str(target_ln).encode("ascii"),
str(target_crc).encode("ascii"),
str(target_n % 1000003).encode("ascii"),
])
target_h = hashlib.sha256(target_payload).hexdigest()
return h == target_h
import math
import hashlib
def to_base36(n):
"""Converts a non-negative integer to its base-36 string representation."""
if n < 0:
raise ValueError("Only non-negative integers can be converted to base-36.")
if n == 0:
return "0"
chars = "0123456789abcdefghijklmnopqrstuvwxyz"
res = ""
while n > 0:
res = chars[n % 36] + res
n //= 36
return res
def is_prime(k):
"""Checks if a number k is prime."""
if k < 2:
return False
for i in range(2, int(math.sqrt(k)) + 1):
if k % i == 0:
return False
return True
def sum_decimal_digits(n):
"""Calculates the sum of the decimal digits of a number n."""
return sum(int(digit) for digit in str(n))
def mystery(x):
"""
This function takes a value x (string) and returns True if it meets
a series of string, numeric, bitwise, and cryptographic criteria.
"""
# Condition 1: x must be a string of exactly 5 alphanumeric characters (base-36 digits).
if not isinstance(x, str):
return False
# Normalize to lowercase for consistency and robust parsing
x_lower = x.lower()
if not (len(x_lower) == 5 and all(c in "0123456789abcdefghijklmnopqrstuvwxyz" for c in x_lower)):
return False
# Convert the base-36 string to an integer n.
n = int(x_lower, 36)
# Condition 2: The integer value of n must be an odd number.
if n % 2 == 0:
return False
# Condition 3: The number n modulo 103 must be 17.
if n % 103 != 17:
return False
# Condition 4: The sum of the decimal digits of n must be a multiple of 5.
if sum_decimal_digits(n) % 5 != 0:
return False
# Condition 5: The number of set bits in n's binary representation must be a prime number.
if not is_prime(n.bit_count()):
return False
# Condition 6: A cryptographic hash of a payload derived from x must meet a prefix requirement.
# The payload is constructed from the normalized string, the integer n,
# its length, and its bit count.
payload = f"{x_lower}|{n}|{len(x_lower)}|{n.bit_count()}".encode('utf-8')
h = hashlib.sha256(payload).hexdigest()
# The SHA256 hash of the payload must start with five leading zeros.
if not h.startswith("00000"):
return False
return True
import unicodedata
import hashlib
# Fixed secret pepper; makes the internal check opaque.
_PEPPER = b"\x8e\x1f\x03mystery\x00pepper\x7f"
def _canon(x):
"""Canonicalize input into a deterministic byte string."""
if isinstance(x, bool):
return None
if isinstance(x, int):
s = str(x)
elif isinstance(x, float):
if not x.is_integer():
return None
s = str(int(x))
elif isinstance(x, str):
# NFKC collapses look-alike Unicode, then trim.
s = unicodedata.normalize("NFKC", x).strip()
else:
return None
return s.encode("utf-8")
def mystery(x):
b = _canon(x)
if b is None:
return False
# Length gate: must be plausible, prevents huge payloads.
if not (1 <= len(b) <= 64):
return False
# Mix in some structure constraints that are non-obvious from the outside.
s = b.decode("utf-8", errors="strict")
if any(ord(ch) < 32 for ch in s): # forbid control chars
return False
# A small arithmetic gate on bytes (annoying to reason about by inspection).
acc = 0
for i, by in enumerate(b, 1):
acc = (acc * 131 + (by ^ (i * 17 & 0xFF))) & 0xFFFFFFFF
if (acc ^ len(b) * 2654435761) & 0xFF != 0x5A:
return False
# Final opaque gate: exact SHA-256 equality against an internal target.
def H(payload: bytes) -> bytes:
return hashlib.sha256(_PEPPER + b"|" + payload).digest()
target = H(b"opensesame") # chosen to ensure solvable
return H(b) == target