Second pass at it - Now for something completely different... 

#!/usr/bin/perl

use strict; # https://www.perlmonks.org/?node_id=11154597
use warnings;

use List::AllUtils qw( first shuffle );
my @in = shuffle <DATA>;
my @out = shift @in;

LOOP:
while( @in )
  {
  for my $in ( 0 .. $#in )
    {
    if( $in[$in] ne $out[0] ) # put at start
      {
      unshift @out, splice @in, $in, 1;
      next LOOP;
      }
    elsif( $in[$in] ne $out[-1] ) # put at end
      {
      push @out, splice @in, $in, 1;
      next LOOP;
      }
    else # put in first middle place it fits
      {
      if( my $pick = first {
        $out[$_ - 1] ne $in[$in] and $out[$_] ne $in[$in] } 1 .. $#out
+ )
        {
        splice @out, $pick, 0, splice @in, $in, 1;
        next LOOP;
        }
      }
    }
  die "FAILED\nin\n @in\nout\n @out";
  }
print @out;

__DATA__
Line 1
Line 2
Line 3
Line 3
Line 3
Line 4
Line 8

[download]

BTW: It only found 240 different orders in 1e7 runs.

In reply to Re: Algorithm RFC: fast (pseudo-)random shuffle with no repetition by tybalt89
in thread Algorithm RFC: fast (pseudo-)random shuffle with no repetition by Anonymous Monk

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.