# ETOOBUSY ðŸš€ minimal blogging for the impatient

# PWC195 - Special Integers

**TL;DR**

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

Iâ€™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 > $n;
++$count;
}
}
}
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
8877690
real 1m42.454s
user 1m42.037s
sys 0m0.181s
$ time raku raku/ch-1.raku 100000000000000
8877690
real 1m49.882s
user 1m49.890s
sys 0m0.128s
```

Well, for some definition of *reasonable*, at least.

Stay safe!

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