What kind of errors are you getting?
I don't think you can set the database by putting it into mysql_connect as a fourth parameter, so use the mysql_select_db function instead. And also you can't put the queries directly into the PHP as well, you have to use the mysql_query function.
Also with the mysql, I'm presuming you're have cust_id set to auto increment since you give it an empty string as a value? If so, just exclude it from the query all together and myself will set it for you as you insert it.
Here is what it should look like (I think):
$connection = mysql_connect("localhost","username","password") or die("message");
mysql_select_db ("database") or die ("message");
$query = "INSERT INTO game (score, email, name) VALUES ('test','if','works')";