# First perfect square in base n with n unique digits

As long as you have the patience, this will work for bases 2 through 36.

Bases 2 through 19 finish quickly, (about 10 seconds on my system), 20 takes a while, 21 is pretty fast, 22 is glacial. 23 through 26 takes several hours.

Use analytical start value filtering based on observations by [Hout](https://rosettacode.org/wiki/User:Hout)++ and [discussion page](https://rosettacode.org/wiki/Talk:First_perfect_square_in_base_N_with_N_unique_digits#Space_compression_and_proof_.3F).

[Try it online!](https://tio.run/##dVTNbtpAEL7zFBPXBDuBlQ1SqkDz0x4i5VBeoGqQA@uwlVmb3XVThMi5fZ1ce@NR@iJ0Zm1jO1UtYS8zO9/8fDOTcZVcHA65FAZ0/gifP95PwXONWHEFV3AXJZr7k06HdLFQ2gz0Oo8UB@9eGnClD9sO4LPawK02kTJoFCeRgV7Y60MvwJc3BMYe6O7@N3uMNB/jGSHJTMR4hmsIA9jCO4hklGyMmGurLHFdlaYGEHchnoSJkgH990p37FsqZB/GhOtROBhsw/aW7mq09YaMoZqtosw727/awxh9tjDdGatwWpCwq0EpYhvQn5@/KvjtUVuFnMax5lSLlZBecYs9KY4uzzDbAsCHQXGatMyLvL6E5yXIV0Sp8F7@1dbGu07xrvhwKz4alWJZpDQfVKkxvUbFnItEyKfJkUhbFKqa4t853gfv4S17Rx/r@izTZ6IJP7WMGgllQcNCFec4VVWIjGFZ6iIiKJqw/WuLSpcCQtWRorY2rv@6Md4MsZkWkETaQC4TrjXasnkqTSSkxhxmNoAy1Ta9RfOfnpateWmPM@h2IZwFQUC/N5QXaZ7b3InVYwmqR0cbcKaoGANs3dlu/1okAx9cfQ3YiF5t6TOV5nLhsSAI/Z124KUFRY9DJtZp6y7wJMo0Xzht55aYBi91s9Aj@Q9j025WUNPwu7NaQoXs1IY6U0Ka2APnE7IB3eFiDN1wpCmvK@gORoF2@ui4T1CNoWryhwNQZ1atnJub8gjn8J@SwMkJ9HpI/65YS80RRmP7ERrmabbBKXbJHfVDUK2q4sZV8cWWWD3arbAdjy5wA/g7nIp8VUZJbx@elyLh1f0lThC1xaQGa02VNSmCQ9LBuaOtCRlXMZ/jji2257MwS5hiZ4p1zqGcNyHBBjsdAzJI1uy0tXOpY2m5kmdaqhAO@7g1FV/nQvEFisORFV@QOM2MSHGhkvi9FV@SWBvFzXzZmRwOfwE)

```perl
#`[

Only search square numbers that have at least N digits;
smaller could not possibly match.

Only bother to use analytics for large N. Finesse takes longer than brute force for small N.

]

unit sub MAIN ($timer = False);

sub first-square (Int $n) {
    my @start = flat '1', '0', (2 ..^ $n)».base: $n;

    if $n > 10 { # analytics
        my $root  = digital-root( @start.join, :base($n) );
        my @roots = (2..$n).map(*²).map: { digital-root($_.base($n), :base($n) ) };
        if $root ∉ @roots {
            my $offset = min(@roots.grep: * > $root ) - $root;
            @start[1+$offset] = $offset ~ @start[1+$offset];
        }
    }

    my $start = @start.join.parse-base($n).sqrt.ceiling;
    my @digits = reverse (^$n)».base: $n;
    my $sq;
    my $now  = now;
    my $time = 0;
    my $sr;
    for $start .. * {
        $sq = .²;
        my $s = $sq.base($n);
        my $f;
        $f = 1 and last unless $s.contains: $_ for @digits;
        if $timer && $n > 19 && $_ %% 1_000_000 {
            $time += now - $now;
            say "N $n:  {$_}² = $sq <$s> : {(now - $now).round(.001)}s" ~
                " : {$time.round(.001)} elapsed";
            $now = now;
        }
        next if $f;
        $sr = $_;
        last
    }
    sprintf( "Base %2d: %13s² == %-30s", $n, $sr.base($n), $sq.base($n) ) ~
        ($timer ?? ($time + now - $now).round(.001) !! '');
}

sub digital-root ($root is copy, :$base = 10) {
    $root = $root.comb.map({:36($_)}).sum.base($base) while $root.chars > 1;
    $root.parse-base($base);
}

say  "First perfect square with N unique digits in base N: ";
say .&first-square for flat
   2 .. 12, # required
  13 .. 16, # optional
  17 .. 19, # stretch
  20, # slow
  21, # pretty fast
  22, # very slow
  23, # don't hold your breath
  24, # slow but not too terrible
  25, # very slow
  26, #   "
;
```

#### Output:

```
First perfect square with N unique digits in base N:
Base  2:            10² == 100
Base  3:            22² == 2101
Base  4:            33² == 3201
Base  5:           243² == 132304
Base  6:           523² == 452013
Base  7:          1431² == 2450361
Base  8:          3344² == 13675420
Base  9:         11642² == 136802574
Base 10:         32043² == 1026753849
Base 11:        111453² == 1240A536789
Base 12:        3966B9² == 124A7B538609
Base 13:       3828943² == 10254773CA86B9
Base 14:       3A9DB7C² == 10269B8C57D3A4
Base 15:      1012B857² == 102597BACE836D4
Base 16:      404A9D9B² == 1025648CFEA37BD9
Base 17:     423F82GA9² == 101246A89CGFB357ED
Base 18:     44B482CAD² == 10236B5F8EG4AD9CH7
Base 19:    1011B55E9A² == 10234DHBG7CI8F6A9E5
Base 20:    49DGIH5D3G² == 1024E7CDI3HB695FJA8G
Base 21:   4C9HE5FE27F² == 1023457DG9HI8J6B6KCEAF
Base 22:   4F94788GJ0F² == 102369FBGDEJ48CHI7LKA5
Base 23:  1011D3EL56MC² == 10234ACEDKG9HM8FBJIL756
Base 24:  4LJ0HDGF0HD3² == 102345B87HFECKJNIGMDLA69
Base 25: 1011E145FHGHM² == 102345DOECKJ6GFB8LIAM7NH9
Base 26: 52K8N53BDM99K² == 1023458LO6IEMKG79FPCHNJDBA
```


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://trizen.gitbook.io/perl6-rosettacode/programming_tasks/f/first_perfect_square_in_base_n_with_n_unique_digits.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.