In the previous section we showed how MICRO-HILLARY can be used to efficiently solve problems in the 15-puzzle and the 24-puzzle domains. Is it possible to use MICRO-HILLARY to efficiently solve general NxN sliding-tile puzzle problems? We have developed a learning algorithm called PARAMETRIC MICRO-HILLARY that uses MICRO-HILLARY in scalable domains. The original MICRO-HILLARY gets a set of basic operators and a well-behaved heuristic function. PARAMETRIC MICRO-HILLARY gets in addition the name of the domain parameter and an initial value for it. The basic operators and the heuristic function can use the domain parameter in their definition. The algorithm sets the parameter to its initial value and calls MICRO-HILLARY. When MICRO-HILLARY returns (due to quiescence), the parameter is increased and MICRO-HILLARY is called again. The algorithm stops when no new macros are added. The algorithm is shown in Figure 8.

The idea behind this algorithm is to extend the strategy of generating training problems of increasing difficulty. MICRO-HILLARY increases the length of the random sequences to generate more complex problems, while PARAMETRIC MICRO-HILLARY also increases the domain parameter.