This study investigated the independent strategy development of 7- to 12-year-old children from the suburbs of Brisbane, Australia in solving a series of novel, two- and three-dimensional combinatorial problems. For each of the problem types, a sequence of five, increasingly complex, strategies was identified from the children's actions and explanations as they manipulated the problem materials. These strategies ranged from inefficient trial-and-error methods to efficient "odometer" procedures. Changes in the children's strategies as they progressed on the problems suggested modifications to their knowledge of the combinatorial domain. Children's knowledge growth was analyzed in terms of their acquisition of a number of domain-specific principles. Such principled knowledge is, in itself, insufficient for problem solution. Children's ability to monitor their actions, detect and correct errors, and recognize problem completion played a crucial role in goal attainment. With experience in problem solution, the older children, in particular, became more aware of the problems' underlying structure and how they could modify their inefficient strategies to produce more effective solution methods. Contains 51 references. (Author/MDH)