math_function_puzzle.py

"""
Goal: try to find a functions f(x) and g(x) such that f(g(x)) is increasing and g(f(x)) is decreasing.

(This program does not solve the puzzle.)
"""

import math

fns = [
    (lambda x: 0, "0"),
    (lambda x: 1, "1"),
    (lambda x: x, "x"),
    (lambda x: x + 1, "x + 1"),
    (lambda x: -x, "-x"),
    (lambda x: 1 - x, "1 - x"),
    (lambda x: abs(x), "abs(x)"),
    (lambda x: x ** 2, "x^2"),
    (lambda x: x ** 3, "x^3"),
    (lambda x: 2 ** x, "2^x"),
    (lambda x: 1 / x, "1/x"),
    (lambda x: 1 / x, "1/x"),
    (lambda x: x ** 0.5, "x^0.5"),
    (lambda x: math.sqrt(x), "sqrt(x)"),
    (lambda x: math.cbrt(x), "cbrt(x)"),
    (lambda x: math.log(x), "log(x)"),
    (lambda x: math.exp(x), "exp(x)"),
    (lambda x: 1 if x > 0 else -1 if x < -1 else 0, "sgn(x)"),
    (lambda x: x ** 2 if x >= 0 else -(x ** 2), "x^2 if x >= 0 else -(x^2)"),
    (lambda x: -(x ** 0.5) if x >= 0 else (-x) ** 0.5, "-(x^0.5) if x >= 0 else (-x)^0.5"),
]

xs = [-2, -1, 0, 1, 2]

def is_increasing(xs):
    for x in xs:
        if x is None or isinstance(x, complex) or math.isnan(x):
            return False
    for i in range(len(xs) - 1):
        if xs[i] >= xs[i + 1]:
            return False
    return True

def is_decreasing(xs):
    for x in xs:
        if x is None or isinstance(x, complex) or math.isnan(x):
            return False
    for i in range(len(xs) - 1):
        if xs[i] <= xs[i + 1]:
            return False
    return True

# g1 = fns[-2][0]
# g2 = fns[-1][0]
# ys = []
# for x in xs:
#     y = g1(g2(x))
#     ys.append(y)
# print(ys)
# ys = []
# for x in xs:
#     y = g2(g1(x))
#     ys.append(y)
# print(ys)


for f1 in fns:
    for f2 in fns:
        ys1 = []
        for x in xs:
            try:
                y = f1[0](f2[0](x))
                ys1.append(y)
            except Exception as e:
                ys1.append(None) # ignore errors
        ys2 = []
        for x in xs:
            try:
                y = f2[0](f1[0](x))
                ys2.append(y)
            except Exception as e:
                ys2.append(None) # ignore errors
        
        if is_increasing(ys1) and is_decreasing(ys2):
            print(f"f(x) = {f1[1]}, g(x) = {f2[1]}")
            print(ys1)
            print(ys2)

        if is_decreasing(ys1) and is_increasing(ys2):
            print(f"f(x) = {f1[1]}, g(x) = {f2[1]}")
            print(ys1)
            print(ys2)