import hashlib
def _is_prime(n: int) -> bool:
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
def mystery(x):
# Accept only strings
if not isinstance(x, str):
return False
# Length must be prime (filters many candidates)
if not _is_prime(len(x)):
return False
# If there's a second character, it must be 'b' (ord 98)
if len(x) > 1 and ord(x[1]) != 98:
return False
# Sum of character codes must match 294
if sum(ord(c) for c in x) != 294:
return False
# Final cryptographic check: SHA-256 must match the chosen digest
target_digest = "ba7816bf8f01cfea414140de5dae2223b00361a396177a9cb410ff61f20015ad"
if hashlib.sha256(x.encode("utf-8")).hexdigest() != target_digest:
return False
return True
import math
def _is_prime(n: int) -> bool:
"""Helper function to check if a number is prime."""
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
def mystery(x) -> bool:
"""
This function takes a value x and returns True if x satisfies a very specific
set of conditions, otherwise False. The goal is to find such an x.
"""
# Condition 1: The input must be a string.
if not isinstance(x, str):
return False
# Condition 2: The string must have a length of exactly 7 characters.
if len(x) != 7:
return False
# Condition 3: The string must be a palindrome (reads the same forwards and backwards).
if x != x[::-1]:
return False
# Condition 4: All characters in the string must be lowercase English letters ('a' through 'z').
if not all('a' <= c <= 'z' for c in x):
return False
# For a 7-character palindrome (c1 c2 c3 c4 c3 c2 c1), the truly unique characters are c1, c2, c3, and c4.
# Let's extract these characters from the first half of the string.
c1, c2, c3, c4 = x[0], x[1], x[2], x[3]
# Condition 5: The four "unique" characters (c1, c2, c3, c4) must themselves be distinct from each other.
if len(set([c1, c2, c3, c4])) != 4:
return False
# Condition 6: The sum of the ASCII (ordinal) values of ALL characters in the string must be 802.
total_ascii_sum = sum(ord(c) for c in x)
if total_ascii_sum != 802:
return False
# Condition 7: The sum of the ASCII (ordinal) values of the *truly unique* characters
# (i.e., the set {c1, c2, c3, c4}) must be 453.
unique_ascii_sum = sum(ord(c) for c in set(x))
if unique_ascii_sum != 453:
return False
# Condition 8: The sum of the ASCII values of the first three characters (c1, c2, c3) must be a prime number.
first_three_sum = ord(c1) + ord(c2) + ord(c3)
if not _is_prime(first_three_sum):
return False
return True
import hashlib
def _is_prime(n: int) -> bool:
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
def mystery(x):
# Must be a string
if not isinstance(x, str):
return False
# Length constraint (filters many candidates)
if len(x) != 5:
return False
# Second character constraint
if len(x) > 1 and x[1] != 'e':
return False
# Sum of ASCII codes of all characters must match this specific total
if sum(ord(c) for c in x) != 532:
return False
# Sum of ASCII values of the first three characters must be prime
if not _is_prime(sum(ord(c) for c in x[:3])):
return False
# Final cryptographic check: SHA-256 digest must match this target
target_digest = "2cf24dba5fb0a30e26e83b2ac5b9e29e1b161e5c1fa7425e73043362938b9824"
if hashlib.sha256(x.encode("utf-8")).hexdigest() != target_digest:
return False
return True
import math
import hashlib
def _is_prime(n: int) -> bool:
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
def _is_perfect_square(n: int) -> bool:
if n < 0:
return False
sqrt_n = int(math.isqrt(n))
return sqrt_n * sqrt_n == n
def mystery(x):
# Condition 1: Input must be a string.
if not isinstance(x, str):
return False
# Condition 2: Length must be exactly 8 characters.
if len(x) != 8:
return False
# Condition 3: All characters must be either a digit ('0'-'9') or a lowercase English vowel ('a','e','i','o','u').
allowed_chars = "0123456789aeiou"
if not all(c in allowed_chars for c in x):
return False
# Condition 4: The first character must be a vowel.
vowels = "aeiou"
if x[0] not in vowels:
return False
# Condition 5: The last character must be a digit.
digits = "0123456789"
if x[7] not in digits:
return False
# Condition 6: The sum of the ordinal (ASCII) values of the first four characters must be a prime number.
sum_first_four_ords = sum(ord(c) for c in x[:4])
if not _is_prime(sum_first_four_ords):
return False
# Condition 7: The sum of the ordinal (ASCII) values of the last four characters must be a perfect square.
sum_last_four_ords = sum(ord(c) for c in x[4:])
if not _is_perfect_square(sum_last_four_ords):
return False
# Condition 8: The sum of ALL ordinal (ASCII) values of the characters must be exactly 507.
total_sum_ords = sum_first_four_ords + sum_last_four_ords
if total_sum_ords != 507:
return False
# Condition 9: The SHA-256 hash of the string must match a specific target digest.
target_digest = "d866a7b7501a357f8841a1362092323e2a05e4664fb971350a221f576e3e5793"
if hashlib.sha256(x.encode("utf-8")).hexdigest() != target_digest:
return False
return True
import hashlib
def mystery(x) -> bool:
# Accept only strings
if not isinstance(x, str):
return False
# Length constraint (filters many candidates)
if len(x) != 4:
return False
# Second character must be 'e'
if x[1] != 'e':
return False
# Sum of ASCII codes of all characters must equal 448
if sum(ord(c) for c in x) != 448:
return False
# Additional check: MD5 digest must match the chosen target
target_digest = "098f6bcd4621d373cade4e832627b4f6"
if hashlib.md5(x.encode("utf-8")).hexdigest() != target_digest:
return False
return True
import hashlib
def mystery(x) -> bool:
# Condition 1: Input must be a string.
if not isinstance(x, str):
return False
# Condition 2: The string must have a length of exactly 6 characters.
if len(x) != 6:
return False
# Condition 3: All characters in the string must be digits ('0' through '9').
if not all(c.isdigit() for c in x):
return False
# Attempt to convert the string to an integer for further numerical checks.
# This try-except block is largely for robustness; Condition 3 should prevent ValueError.
try:
num_x = int(x)
except ValueError:
return False
# Condition 4: The first character, when interpreted as a digit, must be an even number.
# (Note: '0' is considered an even digit).
if int(x[0]) % 2 != 0:
return False
# Condition 5: The last character, when interpreted as a digit, must be an odd number.
if int(x[5]) % 2 != 1:
return False
# Condition 6: The integer value of the entire 6-digit string must be perfectly divisible by 11.
if num_x % 11 != 0:
return False
# Condition 7: The sum of the ordinal (ASCII) values of all six characters must be exactly 292.
total_ord_sum = sum(ord(c) for c in x)
if total_ord_sum != 292:
return False
# Condition 8: The SHA-256 hash of the string must match a specific target digest.
target_digest = "841315b7fb562624d55b08053643734a652a92610b6537a89e0237d6e87f893e"
if hashlib.sha256(x.encode("utf-8")).hexdigest() != target_digest:
return False
return True
def _is_prime(n: int) -> bool:
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
def mystery(x):
# Must be a string of exactly 7 characters
if not isinstance(x, str):
return False
if len(x) != 7:
return False
# All characters must be distinct
if len(set(x)) != 7:
return False
# Must include at least one uppercase, one lowercase, and one digit
if not any(c.isupper() for c in x):
return False
if not any(c.islower() for c in x):
return False
if not any(c.isdigit() for c in x):
return False
# Sum of ASCII codes must equal the chosen target (and be prime)
total = sum(ord(c) for c in x)
if total != 599:
return False
if not _is_prime(total):
return False
# XOR of all character codes must equal the code of the final character
xor_all = 0
for c in x:
xor_all ^= ord(c)
if xor_all != ord(x[-1]):
return False
# Product of character codes modulo a prime must match the chosen residue
MOD = 1013
prod_mod = 1
for c in x:
prod_mod = (prod_mod * ord(c)) % MOD
if prod_mod != 718:
return False
# Sum of squares of ASCII codes must match the chosen total
sum_squares = sum(ord(c) * ord(c) for c in x)
if sum_squares != 55339:
return False
return True
import hashlib
def _is_prime(n: int) -> bool:
"""Helper function to check if a number is prime."""
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
def mystery(x) -> bool:
"""
This function takes a value x and returns True if x satisfies a very specific
set of conditions, otherwise False. The goal is to find such an x.
"""
# Condition 1: The input must be a string.
if not isinstance(x, str):
return False
# Condition 2: The string must have a length of exactly 5 characters.
if len(x) != 5:
return False
# Condition 3: All characters must be lowercase English letters ('a' through 'z').
# Additionally, all characters in the string must be distinct from each other.
if not all('a' <= c <= 'z' for c in x) or len(set(x)) != 5:
return False
# Define vowels and consonants for convenience
vowels = "aeiou"
consonants = "bcdfghjklmnpqrstvwxyz"
# Condition 4: The first character (index 0) must be a vowel.
if x[0] not in vowels:
return False
# Condition 5: The last character (index 4) must be a consonant.
if x[4] not in consonants:
return False
# Condition 6: The ordinal (ASCII) value of the middle character (x[2]) must be a prime number.
if not _is_prime(ord(x[2])):
return False
# Condition 7: The sum of the ordinal (ASCII) values of all five characters must satisfy a specific property.
total_ord_sum = sum(ord(c) for c in x)
# The sum of the digits of this total ordinal sum must be exactly 14.
sum_digits_of_total_sum = sum(int(d) for d in str(total_ord_sum))
if sum_digits_of_total_sum != 14:
return False
# Condition 8: The SHA-256 hash of the string must match a specific target digest.
target_digest = "56c54714659b83e49861619623e1e2474966601b31a89c3683f124c619e917d5"
if hashlib.sha256(x.encode("utf-8")).hexdigest() != target_digest:
return False
return True
import math
from typing import Dict
def _is_probable_prime(n: int) -> bool:
"""Deterministic Miller-Rabin for 32/64-bit integers (sufficient for our range)."""
if n < 2:
return False
small_primes = (2, 3, 5, 7, 11, 13, 17, 19, 23)
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
# witnesses sufficient for 64-bit determinism in practice for our input sizes
witnesses = (2, 3, 5, 7, 11)
def check(a, s, d, n):
x = pow(a, d, n)
if x == 1 or x == n - 1:
return True
for _ in range(s - 1):
x = (x * x) % n
if x == n - 1:
return True
return False
for a in witnesses:
if not check(a, s, d, n):
return False
return True
def mystery(x):
# Must be an integer
if not isinstance(x, int):
return False
# Range constraint (filters many trivial small solutions)
if x < 10**9 or x >= 10**10:
return False
# Must be prime
if not _is_probable_prime(x):
return False
# Sum of decimal digits must be exactly 8
if sum(int(d) for d in str(x)) != 8:
return False
# A set of modular constraints (many coprime moduli -> narrow solution space via CRT)
residues: Dict[int, int] = {
3: 2,
5: 2,
7: 6,
11: 6,
13: 6,
17: 14,
19: 6,
23: 4,
29: 25,
31: 23,
}
for m, r in residues.items():
if x % m != r:
return False
# Last-three-digits constraint
if x % 1000 != 7:
return False
return True
import math
def _is_prime(n: int) -> bool:
"""Helper function to check if a number is prime."""
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
def mystery(x) -> bool:
"""
This function takes an integer x and returns True if x satisfies a very specific
set of numerical and digit-based conditions, otherwise False. The goal is to find such an x.
"""
# Condition 1: The input must be an integer.
if not isinstance(x, int):
return False
# Condition 2: The integer must be an 8-digit number.
# This means x must be between 10,000,000 (10^7) and 99,999,999 (10^8 - 1).
if not (10**7 <= x < 10**8):
return False
# Condition 3: The integer x must be a perfect cube.
# Calculate the cube root and check if it's an integer.
# Using round() helps with potential floating point inaccuracies for perfect cubes.
cbrt_x_float = round(x**(1/3))
if not (cbrt_x_float * cbrt_x_float * cbrt_x_float == x):
return False
# Convert the integer to a string to easily access its digits.
str_x = str(x)
# Condition 4: The string representation of x must contain the digit '3'.
if '3' not in str_x:
return False
# Condition 5: The sum of the digits of x must be a prime number.
sum_digits = sum(int(d) for d in str_x)
if not _is_prime(sum_digits):
return False
# Condition 6: The number of even digits in x must be exactly 5.
# (Even digits are 0, 2, 4, 6, 8. Odd digits are 1, 3, 5, 7, 9.)
even_digit_count = sum(1 for d in str_x if int(d) % 2 == 0)
if even_digit_count != 5:
return False
return True