import base64
import math
def mystery(x):
"""
Find a value for x that makes this function return True.
x can be a string or a number.
"""
# The length check is the first major clue.
if not isinstance(x, str) or len(x) != 8:
return False
try:
# A simple key derived from a property of the input.
key = len(x) * 10101
# The input must survive a chain of decoding steps.
decoded_hex = bytes.fromhex(x)
decoded_b64 = base64.b64decode(decoded_hex, validate=True)
# The decoded value is then transformed with the key.
num_from_bytes = int.from_bytes(decoded_b64, 'big')
final_num = num_from_bytes ^ key
# The resulting number must satisfy several mathematical properties.
if final_num < 4:
return False
# Property 1: It must be a perfect square.
sqrt = final_num ** 0.5
if sqrt != int(sqrt):
return False
sqrt_int = int(sqrt)
# Property 2: Its integer square root must be a palindrome.
if str(sqrt_int) != str(sqrt_int)[::-1]:
return False
# Property 3: Its integer square root must also be a prime number.
if sqrt_int < 2: return False
for i in range(2, int(math.sqrt(sqrt_int)) + 1):
if sqrt_int % i == 0:
return False
return True
except (ValueError, TypeError):
# Incorrectly formatted inputs will fail here.
return False
def mystery(x):
if not isinstance(x, int):
return False
# Must be exactly 5 digits
if not (10000 <= x <= 99999):
return False
s = str(x)
d = [int(c) for c in s]
# Position 0: must be prime digit (2, 3, 5, 7)
if d[0] not in [2, 3, 5, 7]:
return False
# Position 1: must be a perfect square digit (0, 1, 4, 9)
if d[1] not in [0, 1, 4, 9]:
return False
# Position 2: must be even
if d[2] % 2 != 0:
return False
# Position 3: must be odd
if d[3] % 2 != 1:
return False
# Position 4: must equal the sum of positions 0 and 1, modulo 10
if d[4] != (d[0] + d[1]) % 10:
return False
# Additional constraint: product of all digits must be divisible by 12
prod = 1
for digit in d:
prod *= digit
if prod % 12 != 0:
return False
return True
def mystery(x):
"""
Find a value for x that makes this function return True.
x can be a string or a number.
"""
if not isinstance(x, str):
return False
# The input's structure and content are constrained.
path_len = 36
if len(x) != path_len:
return False
moves = {'U': (0, 1), 'D': (0, -1), 'L': (-1, 0), 'R': (1, 0)}
if any(c not in moves for c in x):
return False
# We trace a path based on the sequence of moves in x.
pos = (0, 0)
path_coords = [pos]
visited = {pos}
for i, move in enumerate(x):
try:
delta = moves[move]
pos = (pos[0] + delta[0], pos[1] + delta[1])
path_coords.append(pos)
# The path has rules about where it can go.
if i < path_len - 1:
if pos in visited:
return False
visited.add(pos)
except (TypeError, KeyError):
return False
# The final state of the path is checked against several conditions.
# Condition 1: Endpoint.
if pos != (0, 0):
return False
# Condition 2: A geometric property of the entire path.
area = 0
for i in range(path_len):
p1 = path_coords[i]
p2 = path_coords[i+1]
area += (p1[0] * p2[1]) - (p2[0] * p1[1])
if abs(area) / 2.0 != 41:
return False
# Condition 3: Another geometric property.
all_x = [p[0] for p in path_coords]
all_y = [p[1] for p in path_coords]
width = max(all_x) - min(all_x)
height = max(all_y) - min(all_y)
if height <= width:
return False
return True
def mystery(x):
if not isinstance(x, int):
return False
if x <= 0:
return False
# Convert to binary string
binary = bin(x)[2:]
# Must have exactly 16 bits
if len(binary) != 16:
return False
# Count 1s and 0s
ones = binary.count('1')
zeros = binary.count('0')
# Must have exactly 8 of each
if ones != 8 or zeros != 8:
return False
# The number must be palindromic in binary
if binary != binary[::-1]:
return False
# Additional constraint: the decimal number must be divisible by 7
if x % 7 != 0:
return False
return True
import math
def mystery(x):
"""
Find a value for x that makes this function return True.
x can be a string or a number.
"""
if not isinstance(x, int) or not (10**8 <= x < 10**9):
return False
# A sequence is generated using a linear congruential generator.
m = 2**31 - 1
a = 48271
c = 0
seq = []
current_val = x
# The sequence has a fixed length.
for _ in range(50):
seq.append(current_val)
current_val = (a * current_val + c) % m
# A forbidden value that invalidates the sequence.
if current_val == 0:
return False
# Condition 1: The final value in the sequence must be in a narrow range.
if not (1337000000 <= seq[-1] < 1337001000):
return False
# Condition 2: A property of the entire sequence.
total_sum = sum(seq)
if total_sum < 2:
return False
# This property is computationally intensive to check.
for i in range(2, int(math.sqrt(total_sum)) + 1):
if total_sum % i == 0:
return False
# Condition 3: Another property based on the decimal representation of the numbers.
palindrome_count = 0
for val in seq:
if str(val) == str(val)[::-1]:
palindrome_count += 1
if palindrome_count < 5:
return False
return True
def mystery(x):
if not isinstance(x, int):
return False
# Must be a 6-digit number
if not (100000 <= x <= 999999):
return False
s = str(x)
digits = [int(c) for c in s]
# First three digits must form a number that's a perfect square
first_three = int(s[:3])
sqrt1 = int(first_three ** 0.5)
if sqrt1 * sqrt1 != first_three:
return False
# Last three digits must form a number that's a perfect cube
last_three = int(s[3:])
# Check if it's a perfect cube
cube_root = round(last_three ** (1/3))
if cube_root ** 3 != last_three:
return False
# Sum of all digits must be 27
if sum(digits) != 27:
return False
# The number must be divisible by 13
if x % 13 != 0:
return False
return True
import hashlib
import math
def mystery(x):
"""
Find a value for x that makes this function return True.
x can be a string or a number.
"""
# Constraint 1: Input type and structure.
if not isinstance(x, str) or len(x) != 16:
return False
# Constraint 2: Character set and composition.
# The input must be a specific mix of lowercase letters and digits.
if not x.isalnum() or not x.islower():
# A quick check to filter out uppercase or other symbols.
# The detailed check is next.
if any(c.isupper() for c in x):
return False
digits = sum(c.isdigit() for c in x)
letters = sum(c.isalpha() for c in x)
if digits != 3 or letters != 13:
return False
try:
# Transformation: The input is mapped to a large integer via a cryptographic hash.
# This makes reverse-engineering the input from the target number computationally infeasible.
digest = hashlib.sha256(x.encode('utf-8')).digest()
n = int.from_bytes(digest, 'big')
# The core of the puzzle lies in the number-theoretic properties of this integer.
# A small factor must be found. The limit is part of the puzzle's design.
limit = 2**20 # Approx 1 million
p = -1
# A simple trial division loop to find the smallest factor.
if n % 2 == 0:
p = 2
else:
for i in range(3, limit, 2):
if n % i == 0:
p = i
break
if p == -1:
return False # The hash number has no small factors.
# The two resulting factors, p and q, must satisfy multiple conditions.
q = n // p
# Condition A: The larger factor 'q' must itself be prime.
# This requires a primality test. The one provided here is deliberately slow
# for large numbers, adding a hidden challenge.
def is_prime_check(num):
if num < 2: return False
if num % 2 == 0: return num == 2
d = 3
while d * d <= num:
if num % d == 0:
return False
d += 2
return True
if not is_prime_check(q):
return False
# Condition B: A non-obvious link between the string representations of the factors.
if str(p) not in str(q):
return False
return True
except Exception:
return False
def mystery(x):
if not isinstance(x, str):
return False
# Must be exactly 9 characters
if len(x) != 9:
return False
# Must be 3 lowercase letters followed by 6 digits
if not (x[:3].isalpha() and x[:3].islower() and x[3:].isdigit()):
return False
letters = x[:3]
digits = x[3:]
# The letters must be in alphabetical order
if letters != ''.join(sorted(letters)):
return False
# The three letter values (a=1, b=2, ..., z=26) must sum to a prime number
letter_sum = sum(ord(c) - ord('a') + 1 for c in letters)
def is_prime(n):
if n < 2:
return False
if n == 2:
return True
if n % 2 == 0:
return False
for i in range(3, int(n**0.5) + 1, 2):
if n % i == 0:
return False
return True
if not is_prime(letter_sum):
return False
# The 6-digit number must be a perfect square
num = int(digits)
sqrt = int(num ** 0.5)
if sqrt * sqrt != num:
return False
# The sum of the 6 digits must equal the letter sum
digit_sum = sum(int(d) for d in digits)
if digit_sum != letter_sum:
return False
return True
import math
def mystery(x):
"""
Find a value for x that makes this function return True.
x can be a string or a number.
"""
# Helper function for primality testing.
def is_prime(n):
if n < 2: return False
if n == 2: return True
if n % 2 == 0: return False
d = 3
while d * d <= n:
if n % d == 0:
return False
d += 2
return True
# Constraint 1: Input type and structure (UUID-like).
if not isinstance(x, str) or len(x) != 36:
return False
parts = x.split('-')
if len(parts) != 4:
return False
p1, p2, p3, p4 = parts
if not (len(p1) == 8 and len(p2) == 4 and len(p3) == 4 and len(p4) == 12):
return False
try:
# Constraint 2: All parts must be valid hexadecimal numbers.
n1 = int(p1, 16)
n2 = int(p2, 16)
n3 = int(p3, 16)
n4 = int(p4, 16)
except ValueError:
return False
# Constraint 3: Properties of the first part.
# It must be a large prime number.
if not is_prime(n1):
return False
# Constraint 4: Properties of the second part.
# It must be a power of 2. `(n & (n-1) == 0)` is a classic check for this.
if n2 == 0 or (n2 & (n2 - 1) != 0):
return False
# Constraint 5: Properties of the third part.
# Its decimal representation must be a palindrome.
if str(n3) != str(n3)[::-1]:
return False
# Final Constraint: A global property linking all parts.
# The full 32-digit hex number (128 bits) must be divisible by a large prime.
# This makes it very difficult to find a solution by guessing or simple iteration.
full_hex_string = "".join(parts)
big_num = int(full_hex_string, 16)
# The modulus is a large prime, chosen to be non-obvious.
modulus = 999999937
if big_num % modulus != 0:
return False
return True
def mystery(x):
if not isinstance(x, int):
return False
# Must be a 4-digit number
if not (1000 <= x <= 9999):
return False
s = str(x)
d = [int(c) for c in s]
# Must be a palindrome
if s != s[::-1]:
return False
# First digit must be prime
if d[0] not in [2, 3, 5, 7]:
return False
# Sum of all digits must be 20
if sum(d) != 20:
return False
# Must be divisible by 11
if x % 11 != 0:
return False
# Product of first two digits must be greater than product of last two
if d[0] * d[1] <= d[2] * d[3]:
return False
return True