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 #32.

Find the sum of all products whose multiplicand/multiplier/product identity can be written as a 1 through 9 pandigital.

An example is:

```
39 × 186 = 7254
which contains the numbers 3,9,1,8,6,7,2,5,4 or sorted:
1,2,3,4,5,6,7,8,9
```

My first attempt at brute forcing is to take `{1,2,3,4,5,6,7,8,9}`

, generate all possible permutations and for each permutation squeeze the `x`

and `=`

symbols in between. If we were to start with `{1,2,3,4,5,6,7,8,9}`

, the amount of sums you can make are:

```
1 * 2345678 = 9 F
1 * 234567 = 89 F
1 * 23456 = 789 F
1 * 2345 = 6789 F
1 * 234 = 56789 F
1 * 23 = 456789 F
1 * 2 = 3456789 F
12 * 345678 = 9 F
12 * 34567 = 89 F
12 * 3456 = 789 F
12 * 345 = 6789 F
12 * 34 = 56789 F
12 * 3 = 456789 F
123 * 45678 = 9 F
123 * 4567 = 89 F
123 * 456 = 789 F
123 * 45 = 6789 F
123 * 4 = 56789 F
1234 * 5678 = 9 F
1234 * 567 = 89 F
1234 * 56 = 789 F
1234 * 5 = 6789 F
12345 * 678 = 9 F
12345 * 67 = 89 F
12345 * 6 = 789 F
123456 * 78 = 9 F
123456 * 7 = 89 F
1234567 * 8 = 9 F
```

This means there are 28 variations per permutation multiplied by 9! (362880) which equals to 10.160.640 possibilities. Not only is that a lot, what is also unfortunate is that quite a lot of these 10.160.640 products are false statements. Just checking the 28 I mentioned above, not one of them is correct.

Another brute forcing idea is to start from the highest number you can make, which is: 987.654.321 and determine the divisors of said number. You’ll lower by 1 each time and check if any of the unique divisor pairs and the sum are pandigital 1 till 9. Now, we can already do something a bit smarter here and start from 98.765.432 purely because 987.654.321 will never become pandigital in its product, because all the numbers 1 till 9 are already taken. First up I’ll steal the common divisors method from “Amicable numbers” and tweak it slightly:

```
fn divisors(i: u64) -> Vec<u64> {
let sqrt = (i as f64).sqrt() as u64;
let mut total_div = vec![];
for n in 2..=sqrt {
if i % n == 0 {
if n == i / n {
total_div.push(n);
} else {
total_div.push(n);
total_div.push(i / n);
}
}
}
total_div
}
#[test]
fn test_common_divisors() {
assert_eq!(divisors(60), vec![2, 30, 3, 20, 4, 15, 5, 12, 6, 10]);
assert_eq!(divisors(64), vec![2, 32, 4, 16, 8])
}
```

From this method we can already see very quickly which divisors won’t be pandigital; let’s exclude those by checking for duplicate numbers:

```
fn valid_divisors(i: u64, d1: u64, d2: u64) -> bool {
let mut s = i.to_string();
s.push_str(&d1.to_string());
s.push_str(&d2.to_string());
if s.contains('0') {
return false
}
let vector: Vec<char> = s.chars().collect();
let mut y = vector.clone();
y.sort();
y.dedup();
vector.len() == y.len()
}
#[test]
fn test_valid_divisors() {
assert_eq!(valid_divisors(7254, 39, 186), true);
assert_eq!(valid_divisors(7254, 39, 187), false);
assert_eq!(valid_divisors(7254, 0, 187), false);
}
fn divisors(i: u64) -> Vec<u64> {
let sqrt = (i as f64).sqrt() as u64;
let mut total_div = vec![];
for n in 2..=sqrt {
if i % n == 0 {
if !valid_divisors(i, n, i / n) { continue };
total_div.push(n);
total_div.push(i / n);
}
}
total_div
}
#[test]
fn test_common_divisors() {
assert_eq!(divisors(60), vec![]);
assert_eq!(divisors(7254), vec![39, 186]);
assert_eq!(divisors(64), vec![]);
}
```

The next step would be to start from 98765432 and go down until 0 and see which divisors are listed. This will be slow…

```
fn problem_32() {
let mut n = 98765432;
let mut all_divisors = vec![];
while n > 0 {
let mut d = divisors(n);
if !d.is_empty() {
println!("{}", n);
all_divisors.append(&mut d);
}
n -= 1;
}
println!("{:?}", all_divisors);
}
#[test]
fn test_problem_32() {
problem_32()
}
```

After some fiddling I figured that a good upper bound would be 98765. This is obviously a guess, but it does get the loop started, instead of going from 98765432 (which will take forever). After that code has run (in 2.35s, which is not awful) there’s a list of numbers with all divisors containing numbers consisting of unique digits. Now obviously some of them don’t contain all digits from 1 till 9 (like 6 = 3 x 2), so we’ll filter those out by checking if all of these pairs are in fact pandigital:

```
fn problem_32() -> u64 {
let mut n = 98765;
let mut all_ns: Vec<(u64, u64, u64)> = vec![];
while n > 0 {
let d = divisors(n);
if !d.is_empty() {
for (d1, d2) in &d {
all_ns.push((n, *d1, *d2));
}
}
n -= 1;
}
let mut all_factors = vec![];
for (sum, f1, f2) in all_ns {
let mut string = sum.to_string();
string.push_str(&f1.to_string());
string.push_str(&f2.to_string());
let mut sorted: Vec<char> = string.chars().collect();
sorted.sort();
if sorted == vec!['1','2','3','4','5','6','7','8','9'] {
all_factors.push(f1);
all_factors.push(f2);
}
}
all_factors.iter().sum()
}
#[test]
fn test_problem_32() {
assert_eq!(problem_32(), 5541);
}
```

The sum of all the *factors* is 5541. However, the actual answer is 45228, meaning that I also need to push the totals? If I do that, the number becomes 61911, which is closer, but no cigar. To get the actual answer you require to only add the “total product” and remove the duplicates, which gives me the right answer:

```
fn problem_32() -> u64 {
let mut n = 98765;
let mut all_ns: Vec<(u64, u64, u64)> = vec![];
while n > 0 {
let d = divisors(n);
if !d.is_empty() {
for (d1, d2) in &d {
all_ns.push((n, *d1, *d2));
}
}
n -= 1;
}
let mut all_factors = vec![];
for (sum, f1, f2) in all_ns {
let mut string = sum.to_string();
string.push_str(&f1.to_string());
string.push_str(&f2.to_string());
let mut sorted: Vec<char> = string.chars().collect();
sorted.sort();
if sorted == vec!['1','2','3','4','5','6','7','8','9'] {
all_factors.push(sum);
}
}
all_factors.sort();
all_factors.dedup();
all_factors.iter().sum()
}
#[test]
fn test_problem_32() {
assert_eq!(problem_32(), 45228);
}
```

The code above contains a filter to check for duplicate digits and a filter to check if the number is pandigital. This feels redundant, and we can combine both into one single filter. `valid_divisors`

can be reduced to this:

```
fn valid_divisors(i: u64, d1: u64, d2: u64) -> bool {
let mut s = i.to_string();
s.push_str(&d1.to_string());
s.push_str(&d2.to_string());
let mut vector: Vec<char> = s.chars().collect();
vector.sort();
vector == vec!['1','2','3','4','5','6','7','8','9']
}
```

After implementing the improved `valid_divisors()`

method, the `problem_32()`

-method can be slimmed down quite dramatically. The next thing to do is to see what the highest number in the loop is, which seems to be 7852, so we can start our loop from 7853. With both those changes, we can reduce `problem_32`

down to:

```
fn problem_32() -> u64 {
let mut n = 7853;
let mut all_products: Vec<u64> = vec![];
while n > 0 {
let d = divisors(n);
if !d.is_empty() {
all_products.push(n);
}
n -= 1;
}
all_products.sort();
all_products.dedup();
all_products.iter().sum()
}
#[test]
fn test_problem_32() {
assert_eq!(problem_32(), 45228);
}
```

Talking about refactoring! Obviously the 7853 is a magic number and feels a bit ugly. I feel this can probably be done a lot nicer, but I’ll come back to it at a later point.

Coming back to this code a few days later, I see that 7853 is not a valid upper bound. To revisit this:

- 987654321 is the first upper bound I picked. However, that can never result in pandigital factors, because all the numbers have already been picked once.
- 98765432 was the second candidate. However, splitting that in its factors, is also impossible because no
*two factors*exist that would make this pandigital.

The idea for the upper bound relies on the *two factors*, what are the highest two factors that you can make? If we were to take the group of digits `{1,2,3,4,5,6,7,8,9}`

and start splitting them up in odds and evens, from high to low: `{9,7,5,3,1}`

and `{8,6,4,2}`

. Those would be the highest two factors. However, their product will always contain duplicate digits because all the digits are in use.

To think about it in another way:

```
a * b = c
Where the length needs to be:
al + bl + cl = 9 digits.
What is the max value of c?
```

I can make `c`

9 digits long, but that means `a`

and `b`

need to be 0 digits long, which is impossible. `c`

at least needs to be 7 digits long, so `a`

and `b`

can be 1 digit long, but the highest 1-digit product only gives a 2-digit number (`9 * 8`

). So let’s reduce `c`

down to a 6-digit number, meaning `a`

(or `b`

) will be 2 digits and `b`

(or `a`

) will be 1 digit long. The highest possible product would be (`87 * 9`

), which is a 3-digit number, not a 6-digit number. Let’s repeat this process:

```
a b c
0 0 9 ❌ impossible
1 1 7 ❌ highest product 9 * 8 = 72 (2 digits)
2 1 6 ❌ highest product 87 * 9 = 783 (3 digits)
3 1 5 ❌ highest product 876 * 9 = 7884 (4 digits)
4 1 4 ✅ highest product 8765 * 9 = 78885 (5 digits)
```

`c`

needs to be a 4-digit number, to make two factors of length 1 and length 4, where the highest possible number is 9876.

**
The full solution is available on
GitHub.
**