def mystery(x):
# Obfuscated puzzle: find an integer x that satisfies several conditions.
if not isinstance(x, int):
return False
# Condition 1: x must be prime.
if x < 2:
return False
for i in range(2, int(x**0.5) + 1):
if x % i == 0:
return False
# Condition 2: x must be a palindrome in decimal.
s10 = str(x)
if s10 != s10[::-1]:
return False
# Condition 3: x must be a palindrome in hexadecimal.
s16 = hex(x)[2:]
if s16 != s16[::-1]:
return False
# Condition 4: x must be a palindrome in binary.
s2 = bin(x)[2:]
return s2 == s2[::-1]
def mystery(x):
if not isinstance(x, int):
return False
if x < 0:
return False
root = int(x ** 0.5 + 0.5)
if root * root != x:
return False
s10 = str(x)
if s10 != s10[::-1]:
return False
s16 = hex(x)[2:]
if s16 != s16[::-1]:
return False
s2 = bin(x)[2:]
if s2 != s2[::-1]:
return False
return True
def mystery(x):
# Covert input to integer if it's a string
if isinstance(x, str):
try:
x = int(x)
except:
return False
if not isinstance(x, int) or x < 2:
return False
# Primality test
for d in range(2, int(x ** 0.5) + 1):
if x % d == 0:
return False
# Decimal palindrome
if str(x) != str(x)[::-1]:
return False
# Binary palindrome
b = bin(x)[2:]
if b != b[::-1]:
return False
# Check if x+1 is a perfect cube by exhaustive search
y = x + 1
cube = False
for i in range(1, int(y ** (1/3)) + 2):
if i * i * i == y:
cube = True
break
if not cube:
return False
# Check if x+2 is a perfect square by exhaustive search
z = x + 2
square = False
for j in range(1, int(z ** 0.5) + 2):
if j * j == z:
square = True
break
if not square:
return False
# Check if (x-1)/2 is prime (safe prime condition)
p = (x - 1) // 2
if p < 2:
return False
for d in range(2, int(p ** 0.5) + 1):
if p % d == 0:
return False
# Check if the number of 1's in binary representation is prime
ones = bin(x).count('1')
if ones < 2:
return False
for d in range(2, int(ones ** 0.5) + 1):
if ones % d == 0:
return False
return True
def mystery(x):
if not isinstance(x, int):
return False
if x < 0:
return False
cube = False
max_i = int(x ** (1/3) + 1) + 2
for i in range(0, max_i):
if i * i * i == x:
cube = True
break
if not cube:
return False
s10 = str(x)
if s10 != s10[::-1]:
return False
s16 = hex(x)[2:]
if s16 != s16[::-1]:
return False
s2 = bin(x)[2:]
if s2 != s2[::-1]:
return False
return True
def mystery(x):
# x must be a positive integer
if not isinstance(x, int) or x <= 0:
return False
# Condition 1: primality test
if x < 2:
return False
for i in range(2, int(x**0.5) + 1):
if x % i == 0:
return False
# Condition 2: decimal palindrome
s10 = str(x)
if s10 != s10[::-1]:
return False
# Condition 3: hexadecimal palindrome
s16 = hex(x)[2:]
if s16 != s16[::-1]:
return False
# Condition 4: exactly 5 ones in binary (5 is prime)
b = bin(x)[2:]
if b.count('1') != 5:
return False
# Condition 5: binary length is 17 (17 is prime)
if len(b) != 17:
return False
# Condition 6: congruence modulo 8
if x % 8 != 1:
return False
# Condition 7: product of decimal digits divisible by 144
prod = 1
for ch in s10:
prod *= int(ch)
if prod % 144 != 0:
return False
# Condition 8: contains digit '6'
if '6' not in s10:
return False
# Condition 9: not a binary palindrome
if b == b[::-1]:
return False
return True
def mystery(x):
if isinstance(x, str):
try:
x = int(x)
except ValueError:
return False
if not isinstance(x, int) or x < 2:
return False
# Primality test
for i in range(2, int(x ** 0.5) + 1):
if x % i == 0:
return False
# Decimal palindrome
s = str(x)
if s != s[::-1]:
return False
# Binary palindrome
b = bin(x)[2:]
if b != b[::-1]:
return False
# Sum of digits is prime
sd = sum(int(d) for d in s)
if sd < 2:
return False
for j in range(2, int(sd ** 0.5) + 1):
if sd % j == 0:
return False
# Product of digits is perfect square
prod = 1
for d in s:
prod *= int(d)
root = int(prod ** 0.5 + 0.5)
if root * root != prod:
return False
return True
def mystery(x):
if type(x) == str:
try:
x = int(x)
except:
return False
if not isinstance(x, int) or x < 2:
return False
if x % 2 == 0:
return x == 2
d = 3
while d * d <= x:
if x % d == 0:
return False
d += 2
s = str(x)
if s != s[::-1]:
return False
b = bin(x)[2:]
if b != b[::-1]:
return False
ds = sum(int(c) for c in s)
if ds < 2:
return False
d = 2
while d * d <= ds:
if ds % d == 0:
return False
d += 1
dp = 1
for c in s:
dp *= int(c)
if int(dp ** 0.5) ** 2 != dp:
return False
if x % 19 != 9:
return False
bw = b.count('1')
if bw < 2:
return False
d = 2
while d * d <= bw:
if bw % d == 0:
return False
d += 1
bz = b.count('0')
if int(bz ** 0.5) ** 2 != bz:
return False
if '3' not in hex(x)[2:]:
return False
return True
def mystery(x):
if isinstance(x, str):
try:
x = int(x)
except ValueError:
return False
if not isinstance(x, int) or x < 2:
return False
# Primality test
if x <= 1:
return False
if x <= 3:
is_prime = True
if x % 2 == 0 or x % 3 == 0:
is_prime = False
else:
i = 5
is_prime = True
while i * i <= x:
if x % i == 0 or x % (i + 2) == 0:
is_prime = False
break
i += 6
if not is_prime:
return False
# Decimal palindrome for x
s = str(x)
if s != s[::-1]:
return False
# Binary palindrome
b = bin(x)[2:]
if b != b[::-1]:
return False
# Square is decimal palindrome
sq = x * x
sq_s = str(sq)
if sq_s != sq_s[::-1]:
return False
# Cube is decimal palindrome
cu = sq * x
cu_s = str(cu)
if cu_s != cu_s[::-1]:
return False
return True
def mystery(x):
if isinstance(x, str):
try:
x = int(x)
except:
return False
if not isinstance(x, int) or x < 2:
return False
if x % 2 == 0:
if x == 2:
pass
else:
return False
d = 3
while d * d <= x:
if x % d == 0:
return False
d += 2
s = str(x)
if s != s[::-1]:
return False
b = bin(x)[2:]
if b != b[::-1]:
return False
ds = sum(int(c) for c in s)
if ds < 2:
return False
d2 = 2
while d2 * d2 <= ds:
if ds % d2 == 0:
return False
d2 += 1
prod = 1
for c in s:
prod *= int(c)
if int(prod ** 0.5) ** 2 != prod:
return False
ones = b.count('1')
if ones < 2:
return False
d2 = 2
while d2 * d2 <= ones:
if ones % d2 == 0:
return False
d2 += 1
zeros = b.count('0')
if int(zeros ** 0.5) ** 2 != zeros:
return False
h = hex(x)[2:]
if '3' not in h:
return False
if x % 8 != 1:
return False
if x % 13 != 1:
return False
if x % 19 != 9:
return False
if x % 7 != 5:
return False
ssq = sum(int(c)**2 for c in s)
if ssq < 2:
return False
d2 = 2
while d2 * d2 <= ssq:
if ssq % d2 == 0:
return False
d2 += 1
blen = len(b)
if int(blen ** 0.5) ** 2 != blen:
return False
return True
def mystery(x):
if isinstance(x, str):
try:
x = int(x)
except ValueError:
return False
if not isinstance(x, int) or x < 100:
return False
# Primality test for x >= 100
if x % 2 == 0 or x % 3 == 0:
return False
i = 5
while i * i <= x:
if x % i == 0 or x % (i + 2) == 0:
return False
i += 6
# Decimal palindrome
s = str(x)
if s != s[::-1]:
return False
# Binary palindrome
b = bin(x)[2:]
if b != b[::-1]:
return False
# Sum of digits is prime
sd = sum(int(d) for d in s)
if sd <= 1:
return False
if sd == 2 or sd == 3:
pass
elif sd % 2 == 0 or sd % 3 == 0:
return False
j = 5
while j * j <= sd:
if sd % j == 0 or sd % (j + 2) == 0:
return False
j += 6
# Product of digits is perfect square
prod = 1
for d in s:
prod *= int(d)
root = int(prod ** 0.5 + 0.5)
if root * root != prod:
return False
# Hamming weight (number of 1's in binary) is prime
ones = b.count('1')
if ones <= 1:
return False
if ones == 2 or ones == 3:
pass
elif ones % 2 == 0 or ones % 3 == 0:
return False
k = 5
while k * k <= ones:
if ones % k == 0 or ones % (k + 2) == 0:
return False
k += 6
# Sum of squares of digits is prime
ssq = sum(int(d)**2 for d in s)
if ssq <= 1:
return False
if ssq == 2 or ssq == 3:
pass
elif ssq % 2 == 0 or ssq % 3 == 0:
return False
m = 5
while m * m <= ssq:
if ssq % m == 0 or ssq % (m + 2) == 0:
return False
m += 6
# Hexadecimal representation contains '1'
h = hex(x)[2:]
if '1' not in h:
return False
return True