import math
def mystery(x):
# Path for numbers
if isinstance(x, int):
if x <= 1:
return False
# Helper function to check for primality
def is_prime(n):
if n < 2: return False
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
return False
return True
if not is_prime(x):
return False
# Check specific properties of the prime number
num_set_bits = bin(x).count('1')
# The number must be prime, have exactly 5 set bits in its binary representation,
# and its remainder when divided by 13 must be 7.
return num_set_bits == 5 and (x % 13 == 7)
# Path for strings
elif isinstance(x, str):
# The string must meet several criteria:
# 1. Its length must be exactly 7 characters.
# 2. The sum of the ASCII (ordinal) values of all its characters must be 731.
# 3. The character at index 3 (the fourth character) must be 'd'.
# 4. The greatest common divisor (GCD) of the sum of ASCII values and the string's length must be 1.
if len(x) != 7:
return False
s = sum(ord(c) for c in x)
if s != 731:
return False
if x[3] != 'd':
return False
if math.gcd(s, len(x)) != 1:
return False
return True
# Any other type of input will return False
return False
import hashlib
def mystery(x):
# Two disjoint branches: numbers and strings. Numbers are effectively blocked
# by an almost-impossible composite check; strings must match a specific SHA-1.
if isinstance(x, int):
# Some arbitrary, strict numeric filter that's extremely unlikely to pass.
# (keeps the real solution in the string branch)
if x <= 2:
return False
# require x to be a product of three distinct twin primes larger than 1000
# and also a palindrome in base 7. Very unlikely for typical integers.
def is_prime(n):
if n < 2: return False
if n % 2 == 0:
return n == 2
i = 3
while i * i <= n:
if n % i == 0:
return False
i += 2
return True
# naive factorization up to a modest limit
factors = []
n = x
f = 2
while f * f <= n and len(factors) < 10:
if n % f == 0:
cnt = 0
while n % f == 0:
n //= f
cnt += 1
factors.append((f, cnt))
f += 1 if f == 2 else 2
if n > 1:
factors.append((n, 1))
# require exactly three distinct prime factors, each > 1000 and pairwise twin primes
if len(factors) != 3:
return False
primes = [p for p, c in factors if c == 1 and p > 1000]
if len(primes) != 3:
return False
# check twin-prime condition
for p in primes:
if not (is_prime(p - 2) or is_prime(p + 2)):
return False
# check palindrome in base 7
def is_pal_base(n, base):
s = []
while n:
s.append(n % base)
n //= base
return s == s[::-1]
return is_pal_base(x, 7)
if isinstance(x, str):
# Only short ASCII strings considered
try:
s = x.encode('utf-8')
except Exception:
return False
# length constraint
if len(x) != 5:
return False
# The central check: SHA-1 must match a specific digest.
# (This fixes the valid string(s) to the SHA-1 preimage.)
h = hashlib.sha1(s).hexdigest()
return h == 'aaf4c61ddcc5e8a2dabede0f3b482cd9aea9434d'
# Other types are rejected
return False
import math
def mystery(x):
# Helper functions
def is_prime(n):
if n < 2: return False
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
return False
return True
def is_perfect_square(n):
if n < 0: return False
sqrt_n = int(n**0.5)
return sqrt_n * sqrt_n == n
def get_sum_digits_base10(n):
s = 0
s_str = str(n) # Convert to string to handle 0 easily, or loop with % 10
for digit_char in s_str:
s += int(digit_char)
return s
def get_sum_digits_base(n, base):
if n == 0: return 0
s = 0
while n > 0:
s += (n % base)
n //= base
return s
def is_fibonacci(n):
# A number is Fibonacci if and only if one or both of (5*n^2 + 4) or (5*n^2 - 4) is a perfect square.
if n < 0: return False
if n == 0 or n == 1: return True
return is_perfect_square(5 * n * n + 4) or is_perfect_square(5 * n * n - 4)
# Path for integers
if isinstance(x, int):
# 1. x must be a positive integer, greater than 100.
if not (x > 100):
return False
# 2. x must be a perfect square.
if not is_perfect_square(x):
return False
# 3. The sum of its digits (base 10) must be a Fibonacci number.
sum_digits_x = get_sum_digits_base10(x)
if not is_fibonacci(sum_digits_x):
return False
# 4. When x is converted to base 6, the sum of its digits in base 6 must be a prime number.
sum_base6_digits = get_sum_digits_base(x, 6)
if not is_prime(sum_base6_digits):
return False
# 5. x must contain the digit '3'.
if '3' not in str(x):
return False
return True
# Path for strings
elif isinstance(x, str):
# 1. x must be a string of length 7.
if len(x) != 7:
return False
# Extract digits and letters, ensuring only lowercase English letters or digits
digits_in_x = []
letters_in_x = []
for char in x:
if char.isdigit():
digits_in_x.append(int(char))
elif 'a' <= char <= 'z': # Ensure lowercase English letters only
letters_in_x.append(char)
else: # Contains other characters (e.g., uppercase, symbols)
return False
# 2. x must contain at least one digit and at least one lowercase English letter.
if not (digits_in_x and letters_in_x):
return False
# 3. The sum of the integer values of all digits in x must be a prime number.
sum_of_digits_value = sum(digits_in_x)
if not is_prime(sum_of_digits_value):
return False
# 4. The sum of the ordinal values of all lowercase letters in x must be a perfect square.
sum_of_letter_ord_values = sum(ord(c) for c in letters_in_x)
if not is_perfect_square(sum_of_letter_ord_values):
return False
# 5. The string x must contain the character 'q'.
if 'q' not in x:
return False
# 6. The string x when reversed, must be lexicographically smaller than the original string x.
if x[::-1] >= x:
return False
return True
# Any other type of input will return False
return False
import hashlib
import base64
def mystery(x):
# Reject non-strings quickly
if not isinstance(x, str):
return False
# Must be short printable ASCII
if len(x) != 5:
return False
if any(ord(c) < 32 or ord(c) > 126 for c in x):
return False
# Several intertwined checks (all must hold)
# 1) Sum of ASCII codes
if sum(ord(c) for c in x) != 532:
return False
# 2) Base64 encoding matches a fixed value
if base64.b64encode(x.encode()).decode() != 'aGVsbG8=':
return False
# 3) Known MD5, SHA-1 and SHA-256 digests
if hashlib.md5(x.encode()).hexdigest() != '5d41402abc4b2a76b9719d911017c592':
return False
if hashlib.sha1(x.encode()).hexdigest() != 'aaf4c61ddcc5e8a2dabede0f3b482cd9aea9434d':
return False
if hashlib.sha256(x.encode()).hexdigest() != '2cf24dba5fb0a30e26e83b2ac5b9e29e1b161e5c1fa7425e73043362938b9824':
return False
# 4) Byte-level numeric constraints
v = int.from_bytes(x.encode(), 'big')
# a) specific residue modulo 257 (uses 256 == -1 mod 257 trick)
if v % 257 != 114:
return False
# b) specific popcount
if bin(v).count('1') != 21:
return False
# 5) hex pattern must contain the double-6c sequence
if '6c6c' not in x.encode().hex():
return False
return True
import math
def mystery(x):
# Helper functions
def is_prime(n):
if n < 2: return False
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
return False
return True
def is_perfect_square(n):
if n < 0: return False
sqrt_n = int(n**0.5)
return sqrt_n * sqrt_n == n
def is_fibonacci(n):
# A number is Fibonacci if and only if one or both of (5*n^2 + 4) or (5*n^2 - 4) is a perfect square.
if n < 0: return False
if n == 0 or n == 1: return True
return is_perfect_square(5 * n * n + 4) or is_perfect_square(5 * n * n - 4)
# Path for integers
if isinstance(x, int):
# 1. x must be a positive integer, greater than 100.
if not (x > 100):
return False
# 2. x must be a prime number.
if not is_prime(x):
return False
# 3. The sum of its digits (base 10) must be a Fibonacci number.
sum_digits_x = sum(int(d) for d in str(x))
if not is_fibonacci(sum_digits_x):
return False
# 4. x must contain the digit '1'.
if '1' not in str(x):
return False
# 5. The number of set bits in the binary representation of x must be exactly 5.
if bin(x).count('1') != 5:
return False
return True
# Path for strings
elif isinstance(x, str):
# 1. x must be a string of length 6.
if len(x) != 6:
return False
# 2. x must contain only lowercase English letters.
if not all('a' <= c <= 'z' for c in x):
return False
# 3. x must contain exactly two distinct vowels ('a', 'e', 'i', 'o', 'u').
vowels = {'a', 'e', 'i', 'o', 'u'}
distinct_vowels_in_x = {c for c in x if c in vowels}
if len(distinct_vowels_in_x) != 2:
return False
# 4. The sum of the ordinal values of all characters in x must be a perfect square.
sum_ord_values = sum(ord(c) for c in x)
if not is_perfect_square(sum_ord_values):
return False
# 5. The string x must contain the character 'x' (the letter 'x').
if 'x' not in x:
return False
# 6. The character at index 2 (third character) must be lexicographically smaller
# than the character at index 3 (fourth character).
if not (x[2] < x[3]):
return False
return True
# Any other type of input will return False
return False
import math
import hashlib
def mystery(x):
# Helper: simple deterministic primality (sufficient for moderate sizes)
def is_prime(n):
if n < 2:
return False
if n % 2 == 0:
return n == 2
r = int(math.isqrt(n))
f = 3
while f <= r:
if n % f == 0:
return False
f += 2
return True
# Two very different branches to confuse searchers.
# Numeric branch: a short list of tight arithmetic/bitwise constraints.
if isinstance(x, int):
# Must be a reasonably large positive integer
if x <= 1:
return False
# Must be prime
if not is_prime(x):
return False
# Must be of the form 2^k - 1 (binary all ones)
# property: for such numbers x & (x + 1) == 0
if (x & (x + 1)) != 0:
return False
# Let k be the bit-length (for 2^k - 1 this is k)
k = x.bit_length()
# Exponent k must itself be prime
if not is_prime(k):
return False
# Decimal digit sum must equal 13 (a small prime constraint)
if sum(int(d) for d in str(x)) != 13:
return False
# Fixed modular residue to further narrow candidates
if x % 13 != 5:
return False
return True
# String branch: made intentionally extremely restrictive and effectively
# unreachable to focus the real solution in the numeric branch.
if isinstance(x, str):
# require a specific length
if len(x) != 16:
return False
# require printable ASCII only
if any(ord(c) < 32 or ord(c) > 126 for c in x):
return False
# require a sha256 digest that (for practical purposes) has no known preimage
# (this makes the string branch infeasible to satisfy in practice)
if hashlib.sha256(x.encode('utf-8')).hexdigest() != '0' * 64:
return False
return True
# All other types fail
return False
import math
# Helper functions for the puzzle (defined at module level for efficiency)
def is_prime(n):
if n < 2: return False
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
return False
return True
def is_perfect_square(n):
if n < 0: return False
sqrt_n = int(n**0.5)
return sqrt_n * sqrt_n == n
def get_digits_base(n, base):
if n == 0: return [0]
digits = []
temp_n = n
while temp_n > 0:
digits.append(temp_n % base)
temp_n //= base
return digits[::-1] # Return in correct order (most significant digit first)
def is_palindrome_base(n, base):
digits = get_digits_base(n, base)
return digits == digits[::-1]
def mystery(x):
# Path for integers
if isinstance(x, int):
# 1. x must be a positive integer, greater than 100.
if not (x > 100):
return False
# 2. x must be a prime number.
if not is_prime(x):
return False
# 3. The sum of its digits (base 10) must be a perfect square.
sum_digits_x = sum(int(d) for d in str(x))
if not is_perfect_square(sum_digits_x):
return False
# 4. The binary representation of x must contain at least two '0's.
if bin(x).count('0') < 2:
return False
# 5. x must be a palindrome in base 4.
if not is_palindrome_base(x, 4):
return False
return True
# Path for strings
elif isinstance(x, str):
# 1. x must be a string of length 8.
if len(x) != 8:
return False
# 2. x must contain only lowercase English letters.
if not all('a' <= c <= 'z' for c in x):
return False
# 3. The sum of the ordinal values of all characters in x must be a prime number.
sum_ord_values = sum(ord(c) for c in x)
if not is_prime(sum_ord_values):
return False
# 4. The string must contain exactly three distinct vowels ('a', 'e', 'i', 'o', 'u').
vowels = {'a', 'e', 'i', 'o', 'u'}
distinct_vowels_in_x = {c for c in x if c in vowels}
if len(distinct_vowels_in_x) != 3:
return False
# 5. The character at index 0 must be 'm'.
if x[0] != 'm':
return False
# 6. The string x must be a palindrome when all its vowels are removed.
x_consonants = "".join(c for c in x if c not in vowels)
if x_consonants != x_consonants[::-1]:
return False
return True
# Any other type of input will return False
return False
import math
def _is_probable_prime(n: int) -> bool:
# Deterministic Miller-Rabin for 64-bit integers (works for n < 2^64)
if n < 2:
return False
small_primes = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29)
for p in small_primes:
if n % p == 0:
return n == p
# write n-1 as d*2^s
d = n - 1
s = 0
while d % 2 == 0:
d //= 2
s += 1
# bases sufficient for determinism on 64-bit integers
for a in (2, 325, 9375, 28178, 450775, 9780504, 1795265022):
if a % n == 0:
continue
x = pow(a, d, n)
if x == 1 or x == n - 1:
continue
composite = True
for _ in range(s - 1):
x = (x * x) % n
if x == n - 1:
composite = False
break
if composite:
return False
return True
def mystery(x):
# Numeric branch: designed to be extremely narrow (Mersenne-prime style)
if isinstance(x, int):
if x <= 1:
return False
# must be prime
if not _is_probable_prime(x):
return False
# must be of the form 2^k - 1 (all-ones binary)
if (x & (x + 1)) != 0:
return False
# k is the bit-length (for 2^k - 1 this equals k)
k = x.bit_length()
# exponent k must be prime
if not _is_probable_prime(k):
return False
# decimal digit sum must equal a specific prime (tightens candidates)
if sum(int(d) for d in str(x)) != 73:
return False
# fixed residue modulo 13 to further narrow possibilities
if x % 13 != 1:
return False
# require that the number of 1-bits equals k (sanity for 2^k - 1)
if bin(x).count('1') != k:
return False
return True
# String branch: intentionally infeasible (keeps real solution numeric)
if isinstance(x, str):
# must be short printable ASCII
if len(x) != 16:
return False
if any(ord(c) < 32 or ord(c) > 126 for c in x):
return False
# require an impossible-to-find fixed SHA-256 (practically unreachable)
import hashlib
if hashlib.sha256(x.encode('utf-8')).hexdigest() != '0' * 64:
return False
return True
# other types rejected
return False
import math
import hashlib
# Helper functions for the puzzle
def is_prime(num):
if num < 2: return False
for i in range(2, int(math.sqrt(num)) + 1):
if num % i == 0:
return False
return True
def is_perfect_square(n):
if n < 0: return False
sqrt_n = int(n**0.5)
return sqrt_n * sqrt_n == n
def is_fibonacci(n):
if n < 0: return False
if n == 0 or n == 1: return True
# A number is Fibonacci if and only if one or both of (5*n^2 + 4) or (5*n^2 - 4) is a perfect square.
return is_perfect_square(5 * n * n + 4) or is_perfect_square(5 * n * n - 4)
def get_distinct_prime_factors(num):
factors = set()
d = 2
temp = num
while d * d <= temp:
if temp % d == 0:
factors.add(d)
while temp % d == 0:
temp //= d
d += 1
if temp > 1:
factors.add(temp)
return factors
def get_perfect_power_bases_exponents(num):
"""
Returns a list of (base, exponent) tuples such that num = base^exponent and exponent > 1.
This function handles potential floating point inaccuracies for roots.
"""
if num <= 1:
return []
pairs = []
# Maximum possible exponent k for num = n^k is log2(num)
max_k = int(math.log2(num)) + 1
for k in range(2, max_k):
# Calculate kth root. Using float power and checking nearest integers for robustness.
n_root_approx = num**(1.0/k)
# Check integer candidates around the approximate root
n_candidates = [int(n_root_approx)]
if n_root_approx - int(n_root_approx) > 0.999999999: # Close to the next integer
n_candidates.append(int(n_root_approx) + 1)
elif int(n_root_approx) > 0 and int(n_root_approx)**k > num and n_root_approx - int(n_root_approx) < 0.000000001:
# If current candidate overshoots due to float precision, check previous integer
n_candidates.append(int(n_root_approx) - 1)
for n_candidate in set(n_candidates): # Use set to avoid duplicates
if n_candidate > 1 and n_candidate**k == num:
pairs.append((n_candidate, k))
return pairs
def mystery(x):
# Path for integers
if isinstance(x, int):
# 1. x must be a positive integer, greater than 1.
if x <= 1:
return False
# 2. x must be a palindrome in base 10.
s_x = str(x)
if s_x != s_x[::-1]:
return False
# 3. The sum of its digits (base 10) must be a perfect square.
sum_digits_x = sum(int(d) for d in s_x)
if not is_perfect_square(sum_digits_x):
return False
# 4. x must have exactly two distinct prime factors.
distinct_factors = get_distinct_prime_factors(x)
if len(distinct_factors) != 2:
return False
# 5. x must be a perfect power n^k where k > 1.
# And for at least one such representation (n, k):
# a. k must be a prime number.
# b. n must be a Fibonacci number.
perfect_power_pairs = get_perfect_power_bases_exponents(x)
found_valid_power = False
for n, k in perfect_power_pairs:
if is_prime(k) and is_fibonacci(n):
found_valid_power = True
break
if not found_valid_power:
return False
return True
# Path for strings (Red Herring)
elif isinstance(x, str):
# This path is designed to be practically unreachable by requiring
# a specific SHA-256 hash of a long string.
# 1. String must be of a specific length (e.g., 32 characters)
if len(x) != 32:
return False
# 2. String must contain only printable ASCII characters
if not all(32 <= ord(c) <= 126 for c in x):
return False
# 3. Require a specific SHA-256 digest that has no known preimage
# (practically impossible to find x without brute force)
expected_hash = '0000000000000000000000000000000000000000000000000000000000000000'
if hashlib.sha256(x.encode('utf-8')).hexdigest() != expected_hash:
return False
return True
# Any other type of input will return False
return False
import hashlib
import base64
def mystery(x):
# Numeric branch: intentionally made effectively impossible to satisfy in practice.
# It contains a condition checking for an all-zero SHA-256 of the decimal representation,
# which has no known preimage, so the real solution is in the string branch.
if isinstance(x, int):
if x <= 1:
return False
# quick oddness check
if x % 2 == 0:
return False
# an infeasible cryptographic condition (keeps numeric branch unreachable)
if hashlib.sha256(str(x).encode()).hexdigest() != '0' * 64:
return False
return True
# String branch: multiple tight checks (all must hold)
if isinstance(x, str):
# 1) length constraint
if len(x) != 3:
return False
# 2) lowercase alphabetic only
if not x.isalpha() or not x.islower():
return False
# 3) sum of ASCII values
if sum(ord(c) for c in x) != 294:
return False
# 4) known cryptographic digests (fixes the preimage to a tiny set)
if hashlib.md5(x.encode()).hexdigest() != '900150983cd24fb0d6963f7d28e17f72':
return False
if hashlib.sha1(x.encode()).hexdigest() != 'a9993e364706816aba3e25717850c26c9cd0d89d':
return False
if hashlib.sha256(x.encode()).hexdigest() != 'ba7816bf8f01cfea414140de5dae2223b00361a396177a9cb410ff61f20015ad':
return False
# 5) base64 encoding check (redundant but narrows candidates further)
if base64.b64encode(x.encode()).decode() != 'YWJj':
return False
return True
# Other types rejected
return False