def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules.
It only returns True for a very specific subset of non-negative integers.
"""
if not isinstance(x, int) or x < 0:
return False
try:
s = str(x)
n = len(s)
# First condition
if s != s[::-1]:
return False
# Second condition
if n < 2 or any(n % i == 0 for i in range(2, int(n**0.5) + 1)):
return False
# Third condition
digit_sum = sum(int(d) for d in s)
if digit_sum <= 0 or (digit_sum**0.5) % 1 != 0:
return False
except (ValueError, TypeError):
return False
return True
def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules.
It returns True only for a very specific subset of positive integers.
"""
def is_prime(num):
if num < 2:
return False
for i in range(2, int(num**0.5) + 1):
if num % i == 0:
return False
return True
def prime_factorize(num):
factors = {}
d = 2
temp = num
while d * d <= temp:
while temp % d == 0:
factors[d] = factors.get(d, 0) + 1
temp //= d
d += 1
if temp > 1:
factors[temp] = factors.get(temp, 0) + 1
return factors
if not isinstance(x, int) or x <= 0:
return False
s = str(x)
# Condition 1: Digit-based property
# The product of the digits must be strictly greater than the sum of the digits,
# and their difference must be a prime number.
prod_digits = 1
sum_digits = 0
for digit_char in s:
digit = int(digit_char)
if digit == 0:
# If any digit is 0, the product of digits will be 0.
# This makes prod_digits - sum_digits negative (since sum_digits >= 1 for x > 0),
# which will fail the is_prime check (primes are > 1).
# So, numbers with a '0' digit are implicitly excluded.
prod_digits = 0
break
prod_digits *= digit
sum_digits += digit
if prod_digits <= sum_digits:
return False
diff = prod_digits - sum_digits
if not is_prime(diff):
return False
# Condition 2: Binary representation property
# The binary representation of x must contain the substring "101",
# and the total count of set bits ('1's) must be odd.
bin_x = bin(x)[2:] # Remove "0b" prefix
if "101" not in bin_x:
return False
if bin_x.count('1') % 2 == 0:
return False
# Condition 3: Number theory property ("powerful number")
# x must be a "powerful number", meaning that for every prime factor p of x,
# p^2 must also be a factor of x (i.e., every prime factor has an exponent of at least 2).
factors = prime_factorize(x)
for p, exp in factors.items():
if exp < 2:
return False
return True
import zlib
import math
def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules.
It returns True only for a very specific subset of strings.
"""
if not isinstance(x, str) or len(x) <= 1:
return False
# Condition 1: A special kind of palindrome property
MAPPING = {
'b': 'd', 'd': 'b',
'p': 'q', 'q': 'p',
'n': 'u', 'u': 'n',
'm': 'w', 'w': 'm',
'o': 'o', 's': 's',
'x': 'x', 'z': 'z',
}
try:
# Build the mirrored-and-reversed string
mirrored_reversed = "".join([MAPPING[c] for c in x[::-1]])
if x != mirrored_reversed:
return False
except KeyError:
# x contains characters not in our special alphabet
return False
# Condition 2: A property based on character composition
n_unique = len(set(x))
if n_unique == 0:
return False
# A number k is triangular if and only if 8k + 1 is a perfect square.
val = 8 * n_unique + 1
sqrt_val = int(math.sqrt(val))
if sqrt_val * sqrt_val != val:
return False
# Condition 3: A property based on the string's checksum
try:
checksum = zlib.crc32(x.encode('utf-8'))
if checksum < 2:
return False
# Primality test for the checksum
for i in range(2, int(math.sqrt(checksum)) + 1):
if checksum % i == 0:
return False
except Exception:
return False
return True
import zlib
import math
def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules.
It returns True only for a very specific subset of strings.
"""
if not isinstance(x, str) or len(x) == 0:
return False
# --- Condition 1: Character composition and "semi-palindrome" ---
# The string must contain only alphanumeric characters.
# It must contain at least one digit and at least one letter.
# If all digits are removed (and case ignored), the remaining letters must form a palindrome.
has_digit = False
has_letter = False
alphanum_chars = []
for char in x:
if not char.isalnum():
return False # Only alphanumeric characters allowed
if char.isdigit():
has_digit = True
elif char.isalpha():
has_letter = True
alphanum_chars.append(char)
if not (has_digit and has_letter):
return False
letters_only = "".join([c for c in alphanum_chars if c.isalpha()]).lower()
if letters_only != letters_only[::-1]:
return False
# --- Condition 2: Numerical property from digits ---
# Extract all digits from the string, concatenate them to form an integer N.
# N must be a "powerful number". A number is powerful if for every prime factor p of N, p^2 also divides N.
# (i.e., every prime factor has an exponent of at least 2 in its prime factorization).
digits_str = "".join([c for c in x if c.isdigit()])
if not digits_str: # Should already be caught by has_digit check, but for robustness
return False
try:
n_val = int(digits_str)
except ValueError:
return False
if n_val <= 1: # Powerful numbers are typically defined for positive integers, and 1 is a special case.
return False
def is_powerful(num):
if num <= 1: return False
temp = num
d = 2
while d * d <= temp:
if temp % d == 0:
count = 0
while temp % d == 0:
count += 1
temp //= d
if count < 2: # Exponent must be at least 2
return False
d += 1
if temp > 1: # If a prime factor remains, its exponent was 1, so not powerful.
return False
return True
if not is_powerful(n_val):
return False
# --- Condition 3: Checksum property ---
# The zlib.adler32 checksum of the string (encoded as UTF-8) must have an equal number of '0's and '1's
# in its 32-bit binary representation.
try:
checksum = zlib.adler32(x.encode('utf-8')) & 0xFFFFFFFF # Ensure unsigned 32-bit value
if checksum == 0: # Avoid edge case for 0
return False
# Pad binary representation to 32 bits to correctly count 0s and 1s
bin_checksum = bin(checksum)[2:].zfill(32)
count_zeros = bin_checksum.count('0')
count_ones = bin_checksum.count('1')
if count_zeros != count_ones:
return False
except Exception:
return False
return True
import hashlib
def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules.
It returns True only for a very specific subset of strings.
"""
if not isinstance(x, str) or not x:
return False
# --- Condition 1: Lexical Structure ---
# Must be composed of only lowercase English letters.
if not x.isascii() or not x.islower():
return False
# Must alternate between vowels and consonants.
vowels = "aeiou"
try:
is_vowel_turn = x[0] in vowels
for char in x[1:]:
is_current_char_vowel = char in vowels
if is_current_char_vowel == is_vowel_turn:
return False # Failed to alternate
is_vowel_turn = not is_vowel_turn
except IndexError:
return False # Should be caught by the initial check, but for safety.
# --- Condition 2: Character Composition ---
# The set of unique characters used must satisfy a specific profile.
# It must contain exactly 3 unique vowels and 5 unique consonants.
vowel_chars_present = set()
consonant_chars_present = set()
for char in set(x):
if char in vowels:
vowel_chars_present.add(char)
else:
# We already checked for islower() and isascii(), so no other char types.
consonant_chars_present.add(char)
if len(vowel_chars_present) != 3:
return False
if len(consonant_chars_present) != 5:
return False
# --- Condition 3: Cryptographic Hash Property ---
# The hexadecimal representation of the SHA256 hash of the string
# must end with a sequence of four identical characters.
try:
hex_digest = hashlib.sha256(x.encode('utf-8')).hexdigest()
# Check the last 4 characters
if len(hex_digest) < 4:
return False # Should not happen with SHA256
last_char = hex_digest[-1]
if not (hex_digest[-2] == last_char and \
hex_digest[-3] == last_char and \
hex_digest[-4] == last_char):
return False
except Exception:
# Should not happen for valid strings, but good practice.
return False
return True
import hashlib
import math
from sympy import isprime # Using sympy's isprime for performance in the puzzle itself.
def is_semiprime(num):
"""
Checks if a number is a semi-prime. A semi-prime is a natural number that is the product of exactly two prime numbers,
which may or may not be distinct (e.g., 4=2*2, 6=2*3, 9=3*3, 10=2*5).
"""
if num < 4: # Smallest semiprime is 4 (2*2)
return False
# Find the smallest prime factor (p1)
p1 = -1
d = 2
temp_num = num
# Optimized trial division up to sqrt(num) to find the first factor.
# For very large numbers, this still involves iterating up to sqrt(num),
# but Python's arbitrary-precision integers and sympy's primality test for 'q'
# make this feasible for the numbers involved in this puzzle.
while d * d <= temp_num:
if temp_num % d == 0:
p1 = d
break
d += 1
# If no factor was found (p1 remains -1), it means num itself is prime.
# A prime number is not considered a semiprime (it has only one prime factor).
if p1 == -1:
return False
# Now we have p1, the smallest prime factor of num.
# The number can be expressed as num = p1 * q.
# For num to be a semiprime, q must also be a prime number.
q = num // p1
# Check if q is prime using sympy's optimized primality test.
return isprime(q)
def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules.
It returns True only for a very specific subset of strings.
"""
if not isinstance(x, str) or not x:
return False
# --- Condition 1: Lexical Structure and Character Composition ---
# 1.1 The string must contain only lowercase English letters.
if not x.islower() or not x.isalpha():
return False
# 1.2 The length of the string must be a prime number.
if not isprime(len(x)): # Using sympy.isprime for efficiency
return False
# 1.3 The string must contain exactly 5 unique characters.
unique_chars = sorted(list(set(x)))
if len(unique_chars) != 5:
return False
# 1.4 These 5 unique characters, when sorted alphabetically, must form
# an arithmetic progression based on their ASCII values.
# (i.e., the difference between adjacent character ASCII values is constant).
first_char_ord = ord(unique_chars[0])
second_char_ord = ord(unique_chars[1])
common_difference = second_char_ord - first_char_ord
if common_difference <= 0: # Should not happen if sorted, but for robustness
return False
for i in range(1, len(unique_chars)):
if ord(unique_chars[i]) - ord(unique_chars[i-1]) != common_difference:
return False
# --- Condition 2: Numerical Property from Character Values ---
# Concatenate the ASCII (ordinal) values of all characters in the string x (as strings)
# to form a large integer N. This integer N must be a "semi-prime" number.
try:
n_str_parts = [str(ord(c)) for c in x]
n_val = int("".join(n_str_parts))
if not is_semiprime(n_val):
return False
except (ValueError, TypeError):
# This handles potential issues if ord() or int() conversion fails,
# though with prior checks, it should be robust.
return False
# --- Condition 3: Cryptographic Hash Property ---
# The hexadecimal representation of the SHA1 hash of the string
# (encoded as UTF-8) must start with the sequence "dead".
try:
hex_digest = hashlib.sha1(x.encode('utf-8')).hexdigest()
if not hex_digest.startswith("dead"):
return False
except Exception:
# Catches potential encoding errors or other issues, though unlikely for valid strings.
return False
return True
import uuid
def mystery(x):
"""
This function subjects the input x to a series of rigorous tests.
Only a very specific type of string will pass, combining properties
from graph theory, number theory, and cryptographic hashing.
"""
# --- Preliminary Checks ---
if not isinstance(x, str) or not x.isdigit() or len(x) < 3:
return False
# --- Condition 1: A Graph Traversal Property ---
# The sequence of digits in x must represent a valid "knight's tour"
# on a standard phone keypad.
keypad_moves = {
'1': {'6', '8'}, '2': {'7', '9'}, '3': {'4', '8'},
'4': {'3', '9', '0'}, '5': set(), '6': {'1', '7', '0'},
'7': {'2', '6'}, '8': {'1', '3'}, '9': {'2', '4'},
'0': {'4', '6'}
}
for i in range(len(x) - 1):
current_digit = x[i]
next_digit = x[i+1]
if current_digit not in keypad_moves or next_digit not in keypad_moves[current_digit]:
return False
# --- Condition 2: A Number-Theoretic Property (Smith Number) ---
# The integer represented by x must be a "Smith Number".
# A Smith number is a composite number where the sum of its digits equals
# the sum of the digits of its prime factors.
try:
n = int(x)
def get_factor_digit_sum(num):
factors = []
d = 2
temp = num
while d * d <= temp:
while temp % d == 0:
factors.append(d)
temp //= d
d += 1
if temp > 1:
factors.append(temp)
# A Smith number must be composite (i.e., not prime).
if len(factors) <= 1:
return None
digit_sum = 0
for factor in factors:
# Sum the digits of each prime factor
digit_sum += sum(int(digit) for digit in str(factor))
return digit_sum
sum_of_own_digits = sum(int(digit) for digit in str(n))
sum_of_factor_digits = get_factor_digit_sum(n)
if sum_of_factor_digits is None or sum_of_own_digits != sum_of_factor_digits:
return False
except (ValueError, TypeError):
return False
# --- Condition 3: A Cryptographic Hashing Property ---
# A UUIDv5 is generated from the string x. The number of set bits ('1's)
# in the 128-bit integer representation of this UUID must be a prime number.
try:
def is_prime(num):
if num < 2: return False
# Efficient primality test
for i in range(2, int(num**0.5) + 1):
if num % i == 0: return False
return True
generated_uuid = uuid.uuid5(uuid.NAMESPACE_DNS, x)
popcount = bin(generated_uuid.int).count('1')
if not is_prime(popcount):
return False
except Exception:
return False
return True
import hashlib
import math
def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules.
It returns True only for a very specific subset of strings.
"""
vowels = "aeiou"
# --- Helper functions (defined locally to keep the mystery self-contained) ---
def is_fibonacci(n_val):
if n_val < 1: return False
# A number is Fibonacci if and only if one or both of (5*n^2 + 4) or (5*n^2 - 4) is a perfect square.
# This handles cases where n is 1 or 2 as well.
return math.isqrt(5*n_val*n_val + 4)**2 == (5*n_val*n_val + 4) or \
math.isqrt(5*n_val*n_val - 4)**2 == (5*n_val*n_val - 4)
def is_triangular(n_val):
if n_val < 1: return False
# A number k is triangular if and only if 8k + 1 is a perfect square.
return math.isqrt(8*n_val + 1)**2 == (8*n_val + 1)
def get_positional_value(char):
# 'a' corresponds to 1, 'b' to 2, ..., 'z' to 26
return ord(char) - ord('a') + 1
# --- Preliminary Checks ---
# 1.1 Must be a non-empty string composed of only lowercase English letters.
if not isinstance(x, str) or not x or not x.islower() or not x.isalpha():
return False
# --- Condition 1: Lexical Structure and Palindrome Properties ---
# 1.2 The length of the string must be a Fibonacci number.
if not is_fibonacci(len(x)):
return False
# 1.3 The string must be a palindrome, contain at least two unique characters,
# and its first character must be a vowel while its middle character (x[len(x)//2])
# must be a consonant.
if x != x[::-1]: # Must be a palindrome
return False
if len(set(x)) < 2: # Must contain at least two unique characters
return False
if x[0] not in vowels: # First character must be a vowel
return False
# The middle character (for odd length strings) or one of the two middle characters
# (for even length strings, though this specific check implies odd length for len(x)//2 to be a single char index)
# must be a consonant. Given the palindrome and len(set(x)) >= 2, this also implies that
# x[0] and x[len(x)//2] must be different (vowel vs consonant).
if x[len(x)//2] in vowels: # Middle character must be a consonant
return False
# --- Condition 2: Numerical Property from Character Values ---
# The product of the positional values (a=1, z=26) of all characters in x
# must be a "triangular number".
product_pos_vals = 1
for char in x:
product_pos_vals *= get_positional_value(char)
if not is_triangular(product_pos_vals):
return False
# --- Condition 3: Cryptographic Hash Property ---
# The hexadecimal representation of the SHA1 hash of the string (encoded as UTF-8)
# must have the property that the sum of the numerical values of its first 5
# hexadecimal digits equals the sum of the numerical values of its last 5
# hexadecimal digits. (e.g., 'a' = 10, 'f' = 15).
try:
hex_digest = hashlib.sha1(x.encode('utf-8')).hexdigest()
def hex_digit_sum_segment(hex_str_segment):
total_sum = 0
for h_char in hex_str_segment:
total_sum += int(h_char, 16)
return total_sum
# Check if the hash is long enough (SHA1 is 40 chars, so this is guaranteed)
if len(hex_digest) < 5:
return False
first_five_sum = hex_digit_sum_segment(hex_digest[:5])
last_five_sum = hex_digit_sum_segment(hex_digest[-5:])
if first_five_sum != last_five_sum:
return False
except Exception:
# Catch potential encoding errors or other issues during hashing
return False
return True
import hashlib
import math
def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules,
combining constraints from lexicography, number theory, and cryptography.
It returns True only for a very specific subset of strings.
"""
# --- Preliminary Checks & Condition 1: Lexical Structure ---
if not isinstance(x, str) or not x:
return False
# 1.1: Must be composed of only unique, lowercase English letters.
if not x.islower() or not x.isalpha():
return False
if len(set(x)) != len(x):
return False
# 1.2: The characters must be in strictly increasing alphabetical order.
# This can be checked by comparing the string to its sorted version.
if x != "".join(sorted(x)):
return False
# --- Helper functions for Conditions 2 & 3 ---
def is_perfect(n):
# A number is perfect if it equals the sum of its proper divisors.
# We check against a set of known perfect numbers, which is efficient
# for the scale of numbers this puzzle can generate.
known_perfect_numbers = {6, 28, 496, 8128, 33550336}
return n in known_perfect_numbers
def is_prime(num):
if num < 2:
return False
for i in range(2, int(math.sqrt(num)) + 1):
if num % i == 0:
return False
return True
# --- Condition 2: Number-Theoretic Property ---
# The string is interpreted as a number in base 26, where 'a'=0, 'b'=1, ..., 'z'=25.
# The resulting integer must be a "perfect number".
try:
base26_val = 0
for char in x:
base26_val = base26_val * 26 + (ord(char) - ord('a'))
if not is_perfect(base26_val):
return False
except (ValueError, TypeError):
return False
# --- Condition 3: Cryptographic Hash Property ---
# The sum of the integer values of the hexadecimal digits of the blake2b hash
# of the string (UTF-8 encoded) must be a prime number.
try:
hex_digest = hashlib.blake2b(x.encode('utf-8')).hexdigest()
digit_sum = sum(int(c, 16) for c in hex_digest)
if not is_prime(digit_sum):
return False
except Exception:
return False
return True
import hashlib
import math
def mystery(x):
"""
This function evaluates a value x based on a hidden set of rules,
combining constraints from lexicography, number theory, and cryptography.
It returns True only for a very specific subset of strings.
"""
# --- Helper functions ---
def is_prime(num):
if num < 2: return False
for i in range(2, int(num**0.5) + 1):
if num % i == 0: return False
return True
def is_perfect_square(num):
if num < 0: return False
s = int(math.sqrt(num))
return s*s == num
# --- Preliminary Checks ---
if not isinstance(x, str) or not x:
return False
# --- Condition 1: Lexical Structure and Character Properties ---
# 1.1: Must be composed of only lowercase English letters.
if not x.islower() or not x.isalpha():
return False
# 1.2: The length of the string must be a prime number.
str_len = len(x)
if not is_prime(str_len):
return False
# 1.3: The number of distinct characters in the string must be a perfect square.
num_unique_chars = len(set(x))
if not is_perfect_square(num_unique_chars):
return False
# 1.4: All characters in the string must appear an odd number of times.
# This implies that (sum of counts = string length) and (number of unique characters)
# must both be odd or both be even. Since string length (prime) is always odd (except for 2),
# the number of unique characters must also be odd.
char_counts = {}
for char in x:
char_counts[char] = char_counts.get(char, 0) + 1
for count in char_counts.values():
if count % 2 == 0:
return False
# --- Condition 2: Number-Theoretic Property (Sum of positional values) ---
# The sum of positional values (a=1, b=2, ..., z=26) of all characters in the string
# must be a "highly composite number".
# (A highly composite number is a positive integer that has more divisors than any smaller positive integer).
# We check against a list of known highly composite numbers for efficiency.
known_highly_composite = {1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 1081080, 1441440, 2162160, 2882880, 3603600, 4324320, 7207200, 8648640, 10810800, 14414400, 17297280, 21621600, 32432400, 36036000, 43243200, 64864800, 72072000, 86486400, 108108000, 129729600, 180180000, 216216000, 259459200, 324324000, 432432000, 518918400, 648648000, 778377600, 972972000, 1167561600, 1297296000, 1556755200, 1945944000, 2335132800, 2594592000, 3113510400, 3891888000, 4669065600, 5189184000, 6227020800, 7783776000, 9340531200, 11675616000, 13998960000, 15554400000, 18665280000, 23331600000, 27997920000, 31108800000, 37330560000, 46663200000, 55995840000, 62217600000, 74661120000, 93326400000, 111991680000, 124435200000, 149322240000, 186652800000, 223983360000, 248870400000, 298644480000, 373305600000, 447966720000, 497740800000, 597288960000, 746611200000, 895933440000, 995481600000, 1194577920000, 1493222400000, 1791866880000, 1990963200000, 2389155840000, 2986444800000, 3583733760000, 3981926400000, 4778311680000, 5972889600000, 7167467520000, 7963852800000, 9556623360000, 11945779200000, 14334935040000, 15927705600000, 19113246720000, 23891558400000} # Up to 23.8 trillion
total_pos_val_sum = sum(ord(char) - ord('a') + 1 for char in x)
if total_pos_val_sum not in known_highly_composite:
return False
# --- Condition 3: Cryptographic Hash Property (SHA512) ---
# The SHA512 hash (hexadecimal string) must satisfy two conditions:
# 3.1: The first 10 hexadecimal digits of the hash must contain exactly 'P' prime hexadecimal digits
# (where prime hex digits are 2, 3, 5, 7, b, d) and 'P' itself must be a prime number.
# 3.2: The integer value of the entire SHA512 hash (interpreted as base 16) must be divisible by 13.
try:
hex_digest = hashlib.sha512(x.encode('utf-8')).hexdigest()
# Condition 3.1
prime_hex_chars = {'2', '3', '5', '7', 'b', 'd'}
count_prime_hex_digits = 0
for h_char in hex_digest[:10]: # Only check first 10 characters
if h_char in prime_hex_chars:
count_prime_hex_digits += 1
if not is_prime(count_prime_hex_digits):
return False
# Condition 3.2
hash_int = int(hex_digest, 16)
if hash_int % 13 != 0:
return False
except Exception:
return False
return True