def mystery(x):
    # Input must be a 4-letter alphabetic string
    if not isinstance(x, str) or len(x) != 4:
        return False
    if not x.isalpha():
        return False
    
    # Convert characters to ASCII values
    vals = [ord(c) for c in x]
    a, b, c, d = vals
    
    # A system of equations combining bitwise XOR and arithmetic addition.
    # This combination disrupts standard linear algebra techniques (like Gaussian elimination),
    # requiring a constraint solver, brute force, or careful bitwise analysis.
    
    # Constraint 1: Relationship between 1st and 2nd char
    cond1 = (a ^ b) == 44
    
    # Constraint 2: Relationship between 2nd and 3rd char
    cond2 = (b ^ c) == 21
    
    # Constraint 3: Relationship between 3rd and 4th char
    cond3 = (c ^ d) == 28
    
    # Constraint 4: The arithmetic sum of all characters
    cond4 = (a + b + c + d) == 394
    
    return cond1 and cond2 and cond3 and cond4

def mystery(x):
    # Input must be a 4-character string
    if not isinstance(x, str) or len(x) != 4:
        return False
    
    # Convert characters to their ASCII integer values
    a, b, c, d = [ord(char) for char in x]
    
    # A system of linear constraints
    # These look like random equations, but they form a matrix multiplication
    # M * [a, b, c, d]^T = Target
    
    eq1 = (2*a +   b +  3*c +    d) == 703
    eq2 = (4*a + 3*b +  7*c +  4*d) == 1861
    eq3 = (2*a + 4*b +  9*c +  8*d) == 2518
    eq4 = (8*a + 5*b + 19*c + 13*d) == 4767
    
    return eq1 and eq2 and eq3 and eq4

def mystery(x):
    # Input must be a 5-letter string
    if not isinstance(x, str) or len(x) != 5:
        return False
    
    # Convert characters to ASCII values
    a, b, c, d, e = [ord(char) for char in x]
    
    # A system of non-linear equations combining arithmetic and bitwise operations.
    # This creates a dependency chain that is difficult to invert using standard linear algebra,
    # but solvable via constraint programming (e.g. Z3) or careful algebraic deduction.
    
    # Constraint 1: Non-linear interaction of the first four characters
    # a*b - c*d
    check1 = (a * b - c * d) == -4816
    
    # Constraint 2: Non-linear interaction of the last four characters
    # b*c - d*e
    check2 = (b * c - d * e) == 1636
    
    # Constraint 3: Constraint on the sum of characters at odd indices (0, 2, 4)
    # (a + c + e)^2
    check3 = (a + c + e)**2 == 77284
    
    # Constraint 4: Bitwise mixing of the outer characters and the inner neighbor pair
    # (a XOR e) * (b XOR d)
    check4 = (a ^ e) * (b ^ d) == 1008
    
    # Constraint 5: Simple arithmetic sum of all characters
    check5 = (a + b + c + d + e) == 498
    
    return check1 and check2 and check3 and check4 and check5

def mystery(x):
    if not isinstance(x, str) or len(x) != 6:
        return False
    
    # Extract ASCII values
    v = [ord(c) for c in x]
    
    # Constraint 1: Linear - Sum of values
    if sum(v) != 642:
        return False
    
    # Constraint 2: Linear - Weighted Sum (1-based index)
    if sum((i + 1) * val for i, val in enumerate(v)) != 2301:
        return False
        
    # Constraint 3: Linear - Alternating Sum (+ - + - + -)
    if sum(val * ((-1)**i) for i, val in enumerate(v)) != -28:
        return False
        
    # Constraint 4: Quadratic - Sum of Squares
    if sum(val**2 for val in v) != 69734:
        return False
        
    # Constraint 5: Quadratic - Symmetric Product
    # Products of characters at symmetric positions (0&5, 1&4, 2&3)
    if v[0]*v[5] + v[1]*v[4] + v[2]*v[3] != 34295:
        return False
        
    # Constraint 6: Non-linear - Sum of Pairwise XOR
    # XOR sums of adjacent disjoint pairs
    if (v[0] ^ v[1]) + (v[2] ^ v[3]) + (v[4] ^ v[5]) != 70:
        return False
        
    return True

def mystery(x):
    # Input must be an 8-letter alphabetic string
    if not isinstance(x, str) or len(x) != 8:
        return False
    if not x.isalpha():
        return False
    
    v = [ord(c) for c in x]
    
    # A system of 8 constraints for 8 variables (characters).
    # The system mixes linear algebra, quadratic equations, and bitwise operations.
    # This structure (Mixed Boolean-Arithmetic) resists standard Gaussian elimination 
    # and simple algebraic inversion.
    
    # Constraint 1: Simple Sum
    if sum(v) != 841:
        return False
        
    # Constraint 2: Weighted Sum (0-based index weighting)
    if sum(i * val for i, val in enumerate(v)) != 3006:
        return False
        
    # Constraint 3: Sum of Squares (Quadratic)
    if sum(val**2 for val in v) != 89577:
        return False
        
    # Constraint 4: Alternating Product Sum (Quadratic/Bilinear)
    # v0*v1 - v2*v3 + v4*v5 - v6*v7
    term1 = v[0] * v[1]
    term2 = v[2] * v[3]
    term3 = v[4] * v[5]
    term4 = v[6] * v[7]
    if term1 - term2 + term3 - term4 != -1433:
        return False
        
    # Constraint 5: Product of XOR Sums (Non-linear bitwise mixing)
    # Separates indices into evens (0,2,4,6) and odds (1,3,5,7)
    xor_even = v[0] ^ v[2] ^ v[4] ^ v[6]
    xor_odd = v[1] ^ v[3] ^ v[5] ^ v[7]
    if xor_even * xor_odd != 86:
        return False
        
    # Constraint 6: Linear Symmetry of outer pairs
    # (v0 + v7) - (v1 + v6)
    if (v[0] + v[7]) - (v[1] + v[6]) != -26:
        return False
        
    # Constraint 7: Linear Symmetry of inner pairs
    # (v2 + v5) - (v3 + v4)
    if (v[2] + v[5]) - (v[3] + v[4]) != 13:
        return False
        
    # Constraint 8: Sum of pairwise AND (Bitwise non-linearity)
    # (v0 & v1) + (v2 & v3) + (v4 & v5) + (v6 & v7)
    and_sum = (v[0] & v[1]) + (v[2] & v[3]) + (v[4] & v[5]) + (v[6] & v[7])
    if and_sum != 377:
        return False
        
    return True