解决方法一：使用if语句进行条件判断和变异

# 变异
def mutate_string(s, i, c):
    new_string = s[:i] + c + s[i+1:]
    return new_string

# 跨越
def jump(n):
    if n < 0:
        return "Invalid input"
    elif n == 0:
        return 0
    elif n == 1:
        return 1
    else:
        return jump(n-1) + jump(n-2)

# 条件判断
def check_even_odd(n):
    if n % 2 == 0:
        return "Even"
    else:
        return "Odd"

解决方法二：使用switch语句进行条件判断和变异

# 变异
def mutate_string(s, i, c):
    string_list = list(s)
    string_list[i] = c
    new_string = "".join(string_list)
    return new_string

# 跨越
def jump(n):
    switch = {
        0: 0,
        1: 1,
        2: 1
    }
    if n < 0:
        return "Invalid input"
    elif n in switch:
        return switch[n]
    else:
        return jump(n-1) + jump(n-2)

# 条件判断
def check_even_odd(n):
    switch = {
        0: "Even",
        1: "Odd"
    }
    if n in switch:
        return switch[n]
    else:
        return "Invalid input"

以上是两种基本的解决方法，可以根据具体需求进行调整和扩展。