PWC176 - Permuted Multiples


Here we are with TASK #1 from The Weekly Challenge #176. Enjoy!

The challenge

Write a script to find the smallest integer x such that x, 2x, 3x, 4x, 5x and 6x are permuted multiples of each other.

For example, the integers 125874 and 251748 are permutated multiples of each other as

251784 = 2 x 125874

and also both have the same digits but in different order.



The questions

Just for sake of nitpicking, the solution to the challenge would be no greater than -142857, which is a valid solution. Oh, maybe we’re after positive integer solutions 🤭

The solution

We’ll go brute force, of course, but with a little insight.

The first digit MUST be 1. Anything greater than that would yield one more digit when multiplied by 6, so it would be out of luck.

Moreover, we need a number that is at least six digits long and they must be different from one another. This what happens when you have a leading 1 and you multiply it by 2, 3, and so on up to 6.

Hence, our brute force journey starts at 123456.

We might also note that the maximum number of six digits MUST be below 166667. From that number, a multiplication by 6 yields one more digit, so it’s out of luck.

A loopy Raku first:

#!/usr/bin/env raku
use v6;
sub MAIN {
   my $candidate = 123456;
   loop {
      if check-permuted-multiples-upto6($candidate) {
         put $candidate;

sub check-permuted-multiples-upto6 ($n) {
   my $baseline = $n.comb.Set;
   for 2 .. 6 -> $factor {
      my $candidate = ($n * $factor).comb.Set;
      return False if $candidate (^) $baseline;
   return True;

Sets come to help here: we first build a reference one from the number we’re given as input ($baseline), then one for each multiple ($candidate). To check whether they’re the same or not, we compute the symmetric difference and make sure it’s empty - otherwise digits don’t match and we can move on.

Its translation into Perl cannot leverage Sets, but hashes are pretty cool and we apply mostly the same approach, i.e. build a reference %baseline and check multiples against it.

#!/usr/bin/env perl
use v5.24;
use warnings;
use experimental 'signatures';
no warnings 'experimental::signatures';

my $candidate = 123456;
while ('necessary') {
   if (check_permuted_multiples_upto6($candidate)) {
      say $candidate;

sub check_permuted_multiples_upto6 ($n) {
   my %baseline = map { $_ => 1 } split m{}mxs, $n;
   for my $factor (2 .. 6) {
      for my $digit (split m{}mxs, $n * $factor) {
         return 0 unless exists $baseline{$digit};
   return 1;

Stay safe!

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