Project Euler #62: Cubic permutations

This article is part of a series where I'll be diving head first into the Project Euler puzzles. I want to document the challenge of solving such a puzzle and how I got to the answer. I want to prefix this by stating that I can't cheat for any of these challenges; with that I mean I can't look up any other implementations online. After the implementation, I will validate the answer by using this document or a similar sheet.

In this article I'll be solving: Project Euler #62.


The full text of the puzzle reads the following:

The cube, 41063625 (3453), can be permuted to produce two other cubes: 56623104 (3843) and 66430125 (4053). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.

Find the smallest cube for which exactly five permutations of its digits are cube.

Starting out

I started out with the initial idea of having an infinite loop that keeps on cubing an ever-increasing number n. We take all the permutations of n, and test if they’re cube. If we find 5 cubes, we can stop the loop. The downside to this is “taking all the permutations of n”. A number like 41063625 already has (40320) 8! permutations. That’s quite a lot of calculations to test if just three of these are a cube. This solution is going to be dreadfully slow.

The other idea I had was to keep the infinite loop, but instead if we find a permutation in one of the previous variations of n we would merely keep a list of permutations per key. If we find 5 permutations, stop the loop and print out the first item of the list of permutations.

The code

To keep the list of previous permutations, we would be using a HashMap. The key would merely be the digits of the cubed number sorted as a string. We can use the entry() method on a hash map, to either modify the existing vector, if there is one available, or add a new vector with the cube stored inside of it. We stop the loop once 5 permutations are found, and return the first item (which is going to be the lowest value) of the list of permutations.

let mut list: HashMap<String, Vec<u128>> = HashMap::new();
let mut n: u128 = 1;

loop {
    let cube = n.pow(3);
    let mut r_chars: Vec<char> = cube

    r_chars.sort_by(|a, b| a.cmp(b)); // Sort chars
    let key: String = r_chars.into_iter().collect();

        .and_modify(|c| c.push(cube))

    if let Some(n) = list.get(&key) {
        if n.len() == 5 {
            return n[0]

    n += 1;

The code runs in 0.05s and returns the correct answer of 127035954683.

The full solution is available on GitHub.