use List::Util qw(shuffle);
use strict; use warnings;

my @start = qw( dog cat bird mouse rat snake horse cow pig lizard lamb
+ zebra lion elephant monkey );

my @a = shuffle @start;

for ( 0 .. $#a )
{
    print "@a[ $_, ($_+1)%@a, ($_+3)%@a, ($_+6)%@a ]\n";
}
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The nice thing about this is that it does exactly one shuffle, and never calls rand (except insided the shuffle).