def mystery(x):
# Input handling
if isinstance(x, str):
if x.isdigit():
x = int(x)
else:
return False
if not isinstance(x, int) or x <= 0:
return False
# Primality test
if x < 2:
return False
if x % 2 == 0:
return x == 2
p = 3
while p * p <= x:
if x % p == 0:
return False
p += 2
# Congruence condition
if x % 7 != 1:
return False
# Digit extraction
v1, v2, v3, v4, v5, v6 = 1, 0, 1, 1, 3, 3
cnt = 0
target = x + 1
while True:
if (v1 << 2) + v2 - v3 < v5 * v3:
d = v5
cnt += 1
if cnt == target:
break
v1, v2, v3, v4, v5, v6 = (
10 * v1,
10 * (v2 - d * v3),
v3,
v4,
(10 * (3 * v1 + v2)) // v3 - 10 * d,
v6,
)
else:
v1, v2, v3, v4, v5, v6 = (
v1 * v4,
(2 * v1 + v2) * v6,
v3 * v6,
v4 + 1,
(v1 * (7 * v4 + 2) + v2 * v6) // (v3 * v6),
v6 + 2,
)
return d == 7
def mystery(x):
# Input validation
if not isinstance(x, int):
return False
# Check 64-bit unsigned range
if x < 0 or x >= 18446744073709551616: # 2**64
return False
# Split into two 32-bit halves
left = (x >> 32) & 0xFFFFFFFF
right = x & 0xFFFFFFFF
# Round constants
keys = [0xCAFEBABE, 0xDEADBEEF, 0xBADF00D5, 0x00C0FFEE]
# Run a 4-round Feistel-like network
for key in keys:
# The F-function is simply XOR with the round key
f_val = right ^ key
# Standard Feistel update:
# L_{i+1} = R_i
# R_{i+1} = L_i ^ F(R_i)
new_left = right
new_right = left ^ f_val
# Update state
left, right = new_left, new_right
# The result must be 0
return left == 0 and right == 0
def mystery(x):
# Input validation
if not isinstance(x, str):
return False
if len(x) != 6:
return False
if x[0].isdigit():
return False
# Embedded data (appears random)
s = "chmmyr"
# Compute key from the input string
key = 0
for ch in x:
key ^= ord(ch)
# Transform the key
key = (key * 13 + 17) & 0xFF
# Decode the embedded data using the key
decoded = ''
for ch in s:
decoded += chr(ord(ch) ^ key)
# The input must match the decoded data
return decoded == x
def mystery(x):
# Input validation
if not isinstance(x, str):
return False
try:
# Expecting a specific format of numbers
nums = [int(n) for n in x.split('.')]
except ValueError:
return False
if len(nums) != 4:
return False
# Domain parameters
p = 251
if any(n < 0 or n >= p for n in nums):
return False
# Non-linear layer
# Apply a power map to each element
stage1 = [pow(n, 3, p) for n in nums]
# Linear layer
# Mix the values
total = sum(stage1)
stage2 = [(total + v) % p for v in stage1]
# Target check
return stage2 == [98, 70, 245, 97]
def mystery(x):
# Input validation
if not isinstance(x, int):
return False
if x <= 0:
return False
# Congruence conditions
if x % 2 != 1: return False
if x % 3 != 2: return False
if x % 5 != 1: return False
if x % 7 != 3: return False
if x % 11 != 7: return False
if x % 13 != 5: return False
# Gray code condition: number of ones in gray(x) must be 7
if (x ^ (x >> 1)).bit_count() != 7:
return False
# Primality test
if x < 2:
return False
if x % 2 == 0:
return False
i = 3
while i * i <= x:
if x % i == 0:
return False
i += 2
return True
def mystery(x):
# Input validation
if not isinstance(x, int):
return False
# Ensure 32-bit unsigned integer
if x < 0 or x >= 4294967296: # 2**32
return False
# Generate S-box deterministically
# Affine transformation: S[i] = (137*i + 13) % 256
S = [(i * 137 + 13) % 256 for i in range(256)]
val = x
# Run 2 rounds of a custom SP-network
for _ in range(2):
# 1. Key Mixing
val ^= 0x9E3779B9
# 2. Substitution Layer (S-box)
b0 = S[val & 0xFF]
b1 = S[(val >> 8) & 0xFF]
b2 = S[(val >> 16) & 0xFF]
b3 = S[(val >> 24) & 0xFF]
val = (b3 << 24) | (b2 << 16) | (b1 << 8) | b0
# 3. Permutation Layer (Swap 16-bit halves)
val = ((val << 16) | (val >> 16)) & 0xFFFFFFFF
# 4. Addition with constant
val = (val + 0x12345678) & 0xFFFFFFFF
# The result must be zero
return val == 0
def mystery(x):
# Input validation
if isinstance(x, str):
if x.isdigit():
x = int(x)
else:
return False
if not isinstance(x, int) or x <= 0:
return False
# Condition: x must be odd
if x % 2 == 0:
return False
# Condition: x mod 100 == 29
if x % 100 != 29:
return False
# Condition: x > 1000
if x <= 1000:
return False
# Helper: sum of digits
def digit_sum(n):
s = 0
while n:
s += n % 10
n //= 10
return s
# Harshad check
ds = digit_sum(x)
if x % ds != 0:
return False
# Check sum of two cubes in at least two distinct ways
cube_pairs = 0
a = 1
while a * a * a <= x:
a3 = a * a * a
# Integer cube root of (x - a3)
b = int(round((x - a3) ** (1/3)))
# Adjust b to ensure b^3 <= x-a3 < (b+1)^3
while (b + 1) ** 3 <= x - a3:
b += 1
while b ** 3 > x - a3:
b -= 1
if b >= a and b ** 3 == x - a3:
cube_pairs += 1
if cube_pairs >= 2:
break
a += 1
if cube_pairs < 2:
return False
# Check Carmichael number with exactly 3 prime factors
# First, check composite
if x < 2:
return False
# Simple primality test
def is_prime(n):
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
if is_prime(x):
return False
# Factor x by trial division to get distinct prime factors
n = x
factors = []
i = 2
while i * i <= n:
if n % i == 0:
factors.append(i)
while n % i == 0:
n //= i
i += 1 if i == 2 else 2
if n > 1:
factors.append(n)
# Exactly 3 prime factors
if len(factors) != 3:
return False
# Check square-free
for p in factors:
if x % (p * p) == 0:
return False
# Korselt's criterion
for p in factors:
if (x - 1) % (p - 1) != 0:
return False
return True
def mystery(x):
# Input validation
if not isinstance(x, str) or len(x) != 12:
return False
# Only lowercase english letters
if any(not ('a' <= c <= 'z') for c in x):
return False
# Convert string to list of 0-25 integers
nums = [ord(c) - 97 for c in x]
# Initialization
state = [0, 0, 0, 0]
result = []
# Matrix for linear layer
M = [
[1, 0, 1, 0],
[1, 1, 0, 0],
[0, 1, 1, 1],
[1, 0, 0, 1]
]
# Process in blocks of 4
for i in range(0, 12, 4):
block = nums[i:i+4]
# Step 1: Add previous state (CBC-like chaining)
# Modulo 26 arithmetic
step1 = [(b + s) % 26 for b, s in zip(block, state)]
# Step 2: Affine substitution layer
# f(x) = 5x + 7 (mod 26)
step2 = [(v * 5 + 7) % 26 for v in step1]
# Step 3: Matrix multiplication
step3 = [0] * 4
for r in range(4):
val = 0
for c in range(4):
val += M[r][c] * step2[c]
step3[r] = val % 26
# Update state and result
state = step3
result.extend(step3)
# Check against target sequence
target = [14, 5, 15, 21, 24, 14, 21, 5, 14, 19, 9, 19]
return result == target
def mystery(x):
# Input validation
if not isinstance(x, str):
return False
if len(x) != 10:
return False
if not x.isdigit():
return False
# Convert to digits
digits = [int(c) for c in x]
# Checksum: weighted sum modulo 11
s = sum((i+1) * d for i, d in enumerate(digits))
if s % 11 != 0:
return False
# Split into two 5-digit numbers
first = int(x[:5])
last = int(x[5:])
# Primality test helper
def is_prime(n):
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
return is_prime(first) and is_prime(last)
def mystery(x):
# Input validation
if not isinstance(x, str):
return False
# The path must be exactly 63 steps long (visiting 64 cells)
if len(x) != 63:
return False
# Checksum verification
# ASCII values: 'D'=68, 'U'=85, 'R'=82, 'L'=76
# This acts as a linear constraint on the number of moves of each type.
if sum(ord(c) for c in x) != 4858:
return False
# Simulation of a Self-Avoiding Walk on an 8x8 grid
# Start at top-left (0,0)
r, c = 0, 0
visited = {(r, c)}
for move in x:
if move == 'U': r -= 1
elif move == 'D': r += 1
elif move == 'L': c -= 1
elif move == 'R': c += 1
else: return False
# Check bounds (0-7 for an 8x8 grid)
if not (0 <= r <= 7 and 0 <= c <= 7):
return False
# Path must not intersect itself
if (r, c) in visited:
return False
visited.add((r, c))
# The walk must visit every single cell in the 8x8 grid (Hamiltonian path)
# and must end at the top-right corner (0, 7).
return len(visited) == 64 and (r, c) == (0, 7)