1+2-3*4/5=22 | VnPower

Somebody asked in some channel:

“Given six numbers: 1, 2, 3, 4, 5, 22; and 5 operators: +, -, ×, ÷ and =. Is it possible to arrange these numbers and operators into a statement that is true? You must use all of the numbers and operators given.

Example: 1 + 2 - 3 × 4 ÷ 5 = 22 (not true)”

My approach

SPOILER ALERT!

At first I tried to sort the numbers and operators manually but with no luck.

I wrote a quick Python script to programmatically generate all possible statements and evaluate all of them.

import itertools

def execute(num, ops):
    s = "float("
    ops += " "
    for i in range(6):
        a = num[i]
        b = ops[i]

        s += a + b

        if b == "=":
            s += "float("

    s += ")"
    s = s.replace("6", "22").replace("=", ")==")

    left, right = s.split("==")
    print(s, end="")
    exec(f"print({s}, {left}, {right})")

def iterate():
    ops = '+-*/='
    numbers = '123456'
    perms = list(itertools.permutations(ops))
    perms = [''.join(i) for i in perms]
    nums = list(itertools.permutations(numbers))
    nums = [''.join(i) for i in nums]
    
    for perm in perms:
        for num in nums:
            execute(num, perm)

iterate()

The code is horribly unoptimized and ugly, but at least it gets the job done…

The code prints out a total of 86400 statements. After searching the output dump , it looks like none of them is true.

Afterwords

I wonder if these kinds of problems are possible to mathematically prove that there are no solutions (or otherwise)? If you do came up with a different approach, I would love to see it.