Permutable and

reversible

Weekly challenge 176 — 1 August 2022

Week 176 - 1 Aug 2022

Task 1

Write a script to find the smallest positive 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.

My immediate thought was that the answer is 142857, which I imagine most of us recognise as the repeating decimal sequence in 1/7 (and 2/7, 3/7 ... 6/7). So I could take a punt on a valid solution being: say 142857;

But is there a smaller answer? I spent some time thinking whether I could prove that there isn't, but failed. So I wrote a short script to test it, and indeed that is the right answer.

I also ran the script up to a million and didn't find another answer, so perhaps it is unique?

#!/usr/bin/perl # Peter Campbell Smith - 2022-08-01 # PWC 176 task 1 use v5.28; use utf8; use warnings; my ($j, $j_sorted, $j_length, $k, $jk, ); # loop from 10 to <big> J: for $j (10 .. 1000000) { # sort the digits of $j and get its length $j_sorted = sorted($j); $j_length = length($j); # test 2x, 3x, 4x, 5x, 6x products for $k (2 .. 6) { $jk = $j * $k; # $j is no good if the product has more digits than $j, or they aren't the same digits next J if (length($jk) > $j_length or sorted($jk) ne $j_sorted); } # and if all that passes, we have the answer say qq[Output: $j]; exit; } sub sorted { # sorts the characters of a string return join('', sort(split(//, $_[0]))); }

```
Output: 142857
```

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Peter Campbell Smith is hereby placed in the public domain