def mystery(x):
if not isinstance(x, str):
return False
if len(x) != 10:
return False
if not all('a' <= c <= 'z' for c in x):
return False
if x[0] != 'a':
return False
if x[-1] != 'z':
return False
if x.count('b') != 2:
return False
if 'c' in x:
return False
if sum(ord(c) for c in x) != 1000:
return False
return True
def mystery(x):
if not isinstance(x, int):
return False
if x < 10000 or x > 99999:
return False
# Check if x is a perfect square
sqrt_x = int(x ** 0.5)
if sqrt_x * sqrt_x != x:
return False
# Check if the square root is a palindrome
s = str(sqrt_x)
if s != s[::-1]:
return False
# Sum of digits of x must be divisible by 7
digit_sum = sum(int(d) for d in str(x))
if digit_sum % 7 != 0:
return False
# Product of digits must be greater than 100
product = 1
for d in str(x):
product *= int(d)
if product <= 100:
return False
return True
def mystery(x):
if not isinstance(x, int):
return False
if x <= 0:
return False
n = 2502200483
c = 1622007283
return pow(x, 3, n) == c
def mystery(x):
import hashlib
if not isinstance(x, str):
return False
h = hashlib.sha256(x.encode()).hexdigest()
return h == "2c26b46b68ffc68ff99b453c1d30413413422d706483bfa0f98a5e886266e7ae"
def mystery(x):
# x must be a positive integer
if not isinstance(x, int):
return False
if x <= 0:
return False
# Check if x satisfies the equation x^2 - 1 = 109 * y^2 for some integer y
t = x * x - 1
if t % 109 != 0:
return False
s = t // 109
# Check if s is a perfect square
r = int(s ** 0.5)
return r * r == s
def mystery(x):
if not isinstance(x, int):
return False
if not (100000 <= x <= 999999):
return False
# Check divisibility properties with specific remainders
if x % 37 != 13:
return False
if x % 41 != 29:
return False
if x % 43 != 17:
return False
# Must not contain the digit 0
if '0' in str(x):
return False
return True
def mystery(x):
# x must be an integer
if not isinstance(x, int):
return False
# x must be exactly 4 digits
if not (1000 <= x <= 9999):
return False
# x must be a perfect square
r = int(x ** 0.5)
if r * r != x:
return False
# sum of digits must be a perfect square
s = sum(int(d) for d in str(x))
if int(s ** 0.5) ** 2 != s:
return False
# product of digits must be a perfect cube (allow 0)
p = 1
for d in str(x):
p *= int(d)
if p != 0:
c = int(round(p ** (1/3)))
if c * c * c != p:
return False
# x must be divisible by 17
if x % 17 != 0:
return False
# x must be a Harshad number (divisible by sum of its digits)
if x % s != 0:
return False
# x modulo 19 must be 17
if x % 19 != 17:
return False
# x in octal must not contain the digit '7'
oct_x = oct(x)[2:]
if '7' in oct_x:
return False
# x in hexadecimal must contain at least one letter
hex_x = hex(x)[2:]
if not any(c in 'abcdef' for c in hex_x):
return False
# the reverse of x must not be a perfect square
rev = int(str(x)[::-1])
if int(rev ** 0.5) ** 2 == rev:
return False
# x must not be a palindrome
if str(x) == str(x)[::-1]:
return False
# all digits must be distinct
if len(set(str(x))) != len(str(x)):
return False
# the number of divisors of x must be odd (true for perfect squares)
div_count = 0
for i in range(1, r+1):
if x % i == 0:
div_count += 1
if i != x // i:
div_count += 1
if div_count % 2 == 0:
return False
return True
def mystery(x):
if not isinstance(x, str):
return False
if len(x) != 9:
return False
# Check alternating pattern: letter, digit, letter, digit, ...
for i in range(9):
if i % 2 == 0: # Even positions should be letters
if not x[i].islower():
return False
else: # Odd positions should be digits
if not x[i].isdigit():
return False
# Extract letters and digits
letters = ''.join(x[i] for i in range(0, 9, 2))
digits = [int(x[i]) for i in range(1, 9, 2)]
# Letters must spell a specific word
if letters != 'prime':
return False
# Sum of digits must equal 22
if sum(digits) != 22:
return False
return True
def mystery(x):
# x must be a positive integer
if not isinstance(x, int) or x <= 0:
return False
# Compute the number of Collatz steps to reach 1
def collatz_steps(n):
steps = 0
while n != 1:
if n % 2 == 0:
n //= 2
else:
n = 3 * n + 1
steps += 1
return steps
# The puzzle: find an x for which the Collatz sequence length is exactly 1000
return collatz_steps(x) == 1000
def mystery(x):
if not isinstance(x, int):
return False
if x <= 0:
return False
n = 1000000007
c = 169916223
return pow(x, 3, n) == c