All the course combinations. Well, all the ones the might pan out.

courseLoop.php

/**
 * Creates a list of classes that can be taken (i.e. do not cause conflicts with other classes and can be used
 * in a valid schedule.) from the given courses.
 *
 * Note that this function (the while loop in particular) is a significant performance bottleneck
 *
 * @param ARRAY $courses List of course objects to consider.
 * @return MIXED A list of valid classes or a string with the error message(s).
 */
function findSchedules(array $courses) {
   usort($courses, function($section1, $section2) {
       return count($section1) - count($section2);
   });
   $numCourses = count($courses);
   $indexes = array_fill(0, $numCourses, 0);
   $courseCounts = array();
   foreach($courses as $arr) {
       $courseCounts[] = count($arr);
   }

   $classes = array();
   for($i = 0; $i < $numCourses; $i++) {
       $classes[$i] = $courses[$i][0];
   }
   while(true) {
       if(isValidSchedule($classes)) {
           foreach($classes as $class) {
               $class->conflict = null;
               $class->valid = true;
           }
       }
       //for each course, if the index for this course is less than the max section index, shift it
       //also handles rollover for previous indicies, and updates the currently evaluated classes array
       for($i = 0, $classes[0] = @$courses[0][++$indexes[0]]; $indexes[$i] == $courseCounts[$i];) {
           $classes[$i] = $courses[$i][0];
           $indexes[$i++] = 0;
           //this exits the loop
           if($i == $numCourses) break 2;
           @$classes[$i] = $courses[$i][++$indexes[$i]];
       }
   }

   $conflict = findConflicts($courses);
   if(!empty($conflict)) {
       return implode("<br>", $conflict);
   }
   return $courses;
}