PWC195 - Special Integers


Here we are with TASK #1 from The Weekly Challenge #195. Enjoy!

The challenge

You are given a positive integer, $n > 0.

Write a script to print the count of all special integers between 1 and $n.

An integer is special when all of its digits are unique.

Example 1:

Input: $n = 15
Output: 14 as except 11 all other integers between 1 and 15 are spcial.

Example 2:

Input: $n = 35
Output: 32 as except 11, 22, 33 all others are special.

The questions

I’d probably ask a couple of questions:

  • are we talking about the digits in a decimal representation?
  • is there a limit on $n to consider?

The solution

The number of special integers is clearly limited by the available digits. Hence, the maximum such integer is 9876543210; beyond this, there will always be two digits that are the same (by Pigeonhole principle).

Now, for low values of $n we might opt for a brute force approach. E.g. in Raku we might have:

sub special-integers-bf ($n) {
   my $count = 0;
   for 1 .. $n -> $candidate {
      ++$count if $candidate.comb.Set.elems == $candidate.chars;
   return $count;

The corresponding in Perl:

sub special_integers ($n) {
   my $count = 0;
   for my $candidate (1 .. $n) {
      ++$count if length($candidate) == uniq sort split m{}, $candidate;
   return $count;

Alas, this does not scale well. To be able to count them all (possibly), we have to think differently.

One possible approach is to be generative and only consider permutations over collections of different digits. This is easy to implement in Raku, which comes with permutations and combinations out of the box:

sub special-integers ($n) {
   my $count = 0;
   for 1 .. $n.chars -> $len {
      for combinations([0..9], $len) -> $comb {
         for permutations($comb) -> $perm {
            next if $perm[0] == 0;
            last if $perm.join('').Int > $n;
   return $count;

This, combined with a preliminar check on the input size (capping it at the maximum) gives us something that does computations for any input in a reasonable time:

$ time raku raku/ch-1.raku 9876543210

real	1m42.454s
user	1m42.037s
sys	0m0.181s

$ time raku raku/ch-1.raku 100000000000000

real	1m49.882s
user	1m49.890s
sys	0m0.128s

Well, for some definition of reasonable, at least.

Stay safe!

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