MYSQL RAND() will not generate correct listing

I have a problem getting the right results out of my select. Table wp_rdp_category_images has 3 columns: id (unique) - category_name - category_image.

In the select below I need as result max 4 x category_name and from each category_name, 1 category_image.

Can someone help me correct this select? Would be very nice. Thanks in advance.

Here the php code:

function show_game_category_images_func() {
    $content = '';
    $i = 1;
    $record = $GLOBALS['wpdb']->get_results('
    SELECT * FROM wp_rdp_category_images
    ORDER BY `category_name`, RAND()
    LIMIT 4 
    ');

    if($record){
        foreach($record as $key => $rec){                   
            // if($key == 'picture' && $rec != null && $rec != '')     
            $rec = '<img src="'.$upload_url.$record->category_image.'" width="150" height="150">';

            if ($i == 1) {
                echo '<span><img src="'.$upload_url.$rec->category_image.'" ></span>';  // even
            }
            if ($i == 2) {
                echo '<span><img src="'.$upload_url.$rec->category_image.'" ></span><br>';  // even
            }
            else {
                echo '<span><img src="'.$upload_url.$rec->category_image.'" ></span>';   // odd
            }
            $i++;
        }
    }   
    return $content;
}
add_shortcode('show_game_category_images', 'show_game_category_images_func');

My SQL:

SELECT * FROM `wp_rdp_category_images`    
ORDER BY `category_name`, RAND() LIMIT 4

Here is a test table:

CREATE TABLE IF NOT EXISTS `wp_rdp_category_images` (
  `id` int(12) NOT NULL AUTO_INCREMENT,
  `category_name` varchar(32) NOT NULL,
  `category_image` varchar(64) NOT NULL,
   PRIMARY KEY (`id`)
) ENGINE=InnoDB  DEFAULT CHARSET=latin1 AUTO_INCREMENT=22 ;

INSERT INTO `wp_rdp_category_images` (`id`, `category_name`, `category_image`) VALUES
(1, 'natuur', 'natuur1.jpg'),
(2, 'natuur', 'natuur4.jpg'),
(4, 'aarde', 'aarde3.jpg'),
(5, 'sport', 'sport2.jpg'),
(6, 'aarde', 'aarde3.jpg'),
(7, 'auto', 'autogrijs.jpg'),
(8, 'auto', 'autowit.jpg'),
(9, 'auto', 'autoblack.jpg'),
(10, 'auto', 'autoroodzwart.jpg'),
(11, 'aarde', 'aarde1.jpg'),
(12, 'aarde', 'aarde2.jpg'),
(13, 'sport', 'sport4.jpg'),
(14, 'sport', 'sport3.jpg'),
(15, 'sport', 'sport1.jpg'),
(16, 'natuur', 'natuur2.jpg'),
(17, 'natuur', 'natuur3.jpg'),
(18, 'people', 'people1.jpg'),
(19, 'people', 'people2.jpg'),
(20, 'people', 'people3.jpg'),
(21, 'people', 'people4.jpg');