I rewrote your code and made a stab at my own.

Some of the changes I made to your code:

• No empty cycles without outputing a random number.
• No possibilities of infinite loop.
  • could happen if ($myend/2 < $mystart)
  • example: mystart = 60, myend = 100).
• Actually use int function, not a split on a period
• Use += instead of $sum = $sum + $number

In my code, instead of only choosing numbers from part of the range, I weigh the results smaller or larger as needed. This is an advantage if the desired average is close to the real average, but a disadvantage if the desired average is radically different. The reason I consider it an advantage when the desired average is close to the real average is that you still could get the entire ranger of numbers; not a sequence of numbers below the average, then a sequence of numbers above the average, etc.

use strict;
use warnings;

# Comment out the tests you don't want to run.
my @choices;
@choices = (0, 100, 50, 50);
@choices = (25, 75, 50, 50);
@choices = (0, 100, 50, 10);

my @numbers = his_method(@choices);
#my @numbers = my_method(@choices);

print "Results:\n@numbers\n";
exit;

sub his_method { # re-written
    my $mystart = shift;
    my $myend = shift;
    my $count = shift;
    my $my_average = shift;
    
    my ($start, $end) = ($mystart, $myend);
    my ($number, @numbers);
    my ($sum, $real_average) = 0;

    print "\nList of random values:\n";

    for (my $i = 1; $i <= $count; $i++) {
        $number = int(rand($end+1)) + $start;
        push @numbers, $number;
        $sum += $number;
        $real_average = $sum/$i;
    
            print "Random number $i: $number ($start, $end)\n";
            print "Average: $real_average\n\n";

        ($start, $end) = 
            ($real_average < $my_average) ? # make larger
              ( $my_average, $myend-$my_average)
          : ($real_average > $my_average) ? # make smaller
              ( $mystart, $my_average - $mystart)
          :  # if no adjustment needed.
              ( $mystart, $myend);
    }
    return @numbers;
}

sub my_method {
    my $mystart = shift;
    my $myend = shift;
    my $count = shift;
    my $my_average = shift;

    my ($number, @numbers);
    my ($sum, $real_average) = 0;
    my $next_adj = 1;

    print "\nList of random values:\n";

    for (my $i = 1; $i <= $count; $i++) {
        $number = int(rand()**$next_adj * ($myend+1)) + $mystart;
        push @numbers, $number;
        $sum += $number;
        $real_average = $sum/$i;

            print "Random number $i: $number ($next_adj)\n";
            print "Average: $real_average\n\n";

        $next_adj = 
            ($real_average < $my_average) ? # make larger
              .5 # so take square root of fraction 
          : ($real_average > $my_average) ? # make smaller
              2 # so take power of 2
          : # if no adjustment needed.
              1; 
    }
    return @numbers;
}
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