def mystery(x):
if not isinstance(x, int) or not (1000 <= x < 10000):
return False
p = lambda n: n > 1 and all(n % k for k in range(2, int(n**.5)+1))
import math
q, s = math.isqrt(x), str(x)
return q*q == x and p(q) and p(sum(map(int, s))) and p(int(s[:2])) and p(int(s[2:]))
def mystery(x):
if not isinstance(x, str) or len(x) != 4:
return False
v = list(map(ord, x))
if not all(97 <= i <= 122 for i in v):
return False
# Combined constraints on character values
cond1 = v[0] + v[2] == 200
cond2 = v[1] + v[3] == 213
cond3 = (v[0] ^ v[1] ^ v[2] ^ v[3]) == 19
# Positional bit-shifting and multiplication
a = (v[0] << 8) | v[1]
b = (v[2] << 8) | v[3]
cond4 = (a * 31 - b * 23) == 218963
return cond1 and cond2 and cond3 and cond4
def mystery(x):
import math
if not isinstance(x, int) or x <= 0:
return False
s = str(x)
if len(s) != 6:
return False
def cond_a(n):
for delta in [4, -4]:
v = 5 * n * n + delta
if v > 0:
r = math.isqrt(v)
if r * r == v:
return True
return False
if not cond_a(x):
return False
p = lambda n: n > 1 and all(n % k for k in range(2, int(n**0.5) + 1))
digit_sum = sum(int(c) for c in s)
if not p(digit_sum):
return False
first3 = int(s[:3])
if math.isqrt(first3) ** 2 != first3:
return False
return True
def mystery(x):
if not isinstance(x, str) or len(x) != 8:
return False
# Ensure it's composed of printable ASCII characters
v = [ord(c) for c in x]
if not all(32 <= i < 127 for i in v):
return False
# Property 1: Sum and sum of squares
if sum(v) != 867 or sum(i**2 for i in v) != 95255:
return False
# Property 2: Alternating sum
if sum(v[i] * ((-1)**i) for i in range(8)) != -27:
return False
# Property 3: A rolling hash-like function
res = 0
for i in v:
res = (res * 31 + i) % 1000000007
if res != 455791231:
return False
return True
def mystery(x):
if not isinstance(x, int) or not (10000 <= x <= 99999):
return False
# Palindrome check
n, rev = x, 0
while n:
rev = rev * 10 + n % 10
n //= 10
if x != rev:
return False
# Primality check
p = lambda n: n > 1 and all(n % k for k in range(2, int(n**0.5)+1))
if not p(x):
return False
# Digit condition
s = str(x)
ds = sum(int(c) for c in s)
if ds * (ds + 1) != 182:
return False
# Binary condition
bc = bin(x).count('1')
if (bc - 3) ** 3 != 64:
return False
return True
def mystery(x):
if not isinstance(x, str) or len(x) != 8:
return False
# Get ASCII values of characters and check their range
v = [ord(c) for c in x]
if not all(33 <= i <= 126 for i in v):
return False
# Condition 1: A weighted sum using powers of 2 as coefficients
c1 = sum(v[i] * (2**i) for i in range(8)) == 20896
# Condition 2: XOR sum of all character codes
xor_res = 0
for b in v:
xor_res ^= b
c2 = xor_res == 114
# Condition 3: Sum of character codes cubed modulo a large prime
c3 = sum(pow(i, 3, 1000003) for i in v) % 1000003 == 976453
# Condition 4: Sum of products of symmetric pairs (v[0]*v[7], v[1]*v[6], etc.)
c4 = sum(v[i] * v[7-i] for i in range(4)) == 21026
# Condition 5: Sum of character codes at even indices
c5 = sum(v[i] for i in range(0, 8, 2)) == 268
return c1 and c2 and c3 and c4 and c5
def mystery(x):
if not isinstance(x, int) or not (10000 <= x <= 99999):
return False
s = str(x)
d = [int(c) for c in s]
import math
# Check 1: x is a perfect square
q = math.isqrt(x)
if q * q != x:
return False
# Check 2: alternating digit sum is divisible by 11
if (d[0] - d[1] + d[2] - d[3] + d[4]) % 11 != 0:
return False
# Check 3: digit sum is itself a perfect square
ds = sum(d)
if math.isqrt(ds) ** 2 != ds:
return False
# Check 4: first digit plus last digit equals middle digit
if d[0] + d[4] != d[2]:
return False
# Check 5: the integer square root of x is divisible by 16
if q % 16 != 0:
return False
return True
def mystery(x):
if not isinstance(x, str) or len(x) != 8:
return False
# Convert characters to their ASCII integer values
v = [ord(c) for c in x]
# Ensure all characters are within the printable ASCII range
if not all(33 <= i <= 126 for i in v):
return False
# Property 1: Statistical sums of character codes
# This involves the total sum, a weighted sum, and a sum of squares
c1 = sum(v) == 663
c2 = sum(i * v[i] for i in range(8)) == 1944
c3 = sum(val**2 for val in v) == 63407
# Property 2: Bitwise XOR sum of all character codes
xor_val = 0
for b in v:
xor_val ^= b
c4 = xor_val == 121
# Property 3: A rolling hash with multiplier 31 and prime modulus 1,000,000,007
h = 0
for b in v:
h = (h * 31 + b) % 1000000007
c5 = h == 815623096
# Property 4: A relationship between products of character pairs
c6 = (v[0]*v[1] - v[2]*v[3] + v[4]*v[5] - v[6]*v[7]) == 1213
return all([c1, c2, c3, c4, c5, c6])
def mystery(x):
if not isinstance(x, int) or not (100000 <= x <= 999999):
return False
import math
s = str(x)
d = [int(c) for c in s]
# Check 1: x is a perfect square
q = math.isqrt(x)
if q * q != x:
return False
# Check 2: Every digit of the square root appears somewhere in x
sq_s = str(q)
for c in sq_s:
if c not in s:
return False
# Check 3: The digit sum of the square root equals the number of digits in x
q_ds = sum(int(c) for c in sq_s)
if q_ds != len(s):
return False
# Check 4: x is a Harshad number (divisible by the sum of its own digits)
ds = sum(d)
if x % ds != 0:
return False
return True
def mystery(x):
if not isinstance(x, str) or len(x) != 8:
return False
v = [ord(c) for c in x]
if not all(32 < i < 127 for i in v):
return False
# Constraint 1: Linear invariants based on character values
s1 = sum(v)
s2 = sum((i + 1) * v[i] for i in range(8))
if s1 != 867 or s2 != 3932:
return False
# Constraint 2: Positional weighting using cubic powers
s3 = sum(v[i] * (i**3) for i in range(8))
if s3 != 83513:
return False
# Constraint 3: A cumulative XOR-addition mixing chain
res = 0
for i, val in enumerate(v):
res = (res ^ val) + i
if res != 51:
return False
# Constraint 4: Alternating products of symmetric pairs (v0, v7), (v1, v6), etc.
if v[0]*v[7] - v[1]*v[6] + v[2]*v[5] - v[3]*v[4] != -4996:
return False
# Constraint 5: Bitwise intersection and internal repetition
if (v[0] & v[1] & v[2]) != 80 or v[2] != v[3]:
return False
# Constraint 6: Fixed difference between start and end character sums
if v[0] + v[1] - v[6] - v[7] != -16:
return False
return True