The following returns all the sets of 5 uniqe letters that agree with the rules (two and only two letters are common between the set of letters and each of the words). Note that I've also added a rule, that every set of letters must contain ONLY letters included in the input words. 
use strict;
use warnings;

my (%l, @a, $bad, $w);
my @w = ('bumps','seams','domes','shake','pokes','dukes');

for (@w) { $l{$_} = () for (split //, $_); }
permute(5, '', \@a, sort keys %l);

for $w (@a) {
    $bad = 0;
    for (@w) {
        if (both($_, $w) != 2) { $bad = 1; last; }
    }
    if (!$bad) { print "$w\n"; }
}

sub permute {
    my ($d, $w, $a, @l) = @_;
    if (!--$d) { push @$a, $w.$_ for @l; }
    else { permute($d, $w.$l[$_], $a, @l[($_+1)..$#l]) for (0..($#l-$d
+)); }
}

sub both {
    my ($w1, $w2) = @_; my %l; my $c = 0;
    $l{$_} = () for split //, $w1;
    for (split //, $w2) { $l{$_} = 1 if exists $l{$_}; }
    for (values %l) { $c++ if $_; }
    return $c;
}

[download]
Sloppy code in places probably, but it does the job. Returned values are: 
abdep
abeou
adkmp
akmou

[download]
Problem is, I can't see a way to arrange any of these into an english word, unless I'm missing a possibility on the first set. Obviously, some of the letters have to be used twice, or letters not in any of the original words have to be allowed. Back to the drawing board :\

EDIT: Ohhh, you changed the puzzle. Time to see if I can get this one to work...

Nope, doesn't work with the new puzzle either. I'd delete my post at this point, but I should leave it here in case someone can gain insight from my mistakes.

In reply to Re: Challenge: Mystery Word Puzzle by TedPride
in thread Challenge: Mystery Word Puzzle by Limbic~Region

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.