Here's another approach. It may not be cleaner, but it's closer to the Fisher-Yates shuffle recommended in the FAQ.

#!/usr/bin/perl -w
use strict;

my @set = (1 .. 9);
my %point;
my ($x, $y);

# something for the last element left to point to
my $last = $set[-1];

# prime the pump
$x = pop @set;

while (@set) {
    # grab something that's left
    $y = splice @set, ((rand @set) - 1), 1;
    $point{$x} = $y;
    # do that one next
    $x = $y;
}

# take care of that straggler
$point{$x} = $last;

# and now a demo
foreach my $key (keys %point) {
    print "$key points to ", $point{$key}, "\n";
}
[download]