- Optimal planning and heuristic search systems solve state-space search problems by finding a least-cost path from start to goal. As a byproduct of having an optimal path they also determine the optimal solution cost. In this paper we focus on the problem of determining the optimal solution cost for a state-space search problem directly, i.e., without actually finding a solution path of that cost. We present an algorithm, BiSS, which is a hybrid of bidirectional search and stratified sampling that produces accurate estimates of the optimal solution cost. BiSS is guaranteed to return the optimal solution cost in the limit as the sample size goes to infinity. We show empirically that BiSS produces accurate predictions in several domains. In addition, we show that BiSS scales to state spaces much larger than can be solved optimally. In particular, we estimate the average solution cost for the 6 × 6 , 7 × 7 , and 8 × 8 Sliding-Tile puzzle and provide indirect evidence that these estimates are accurate. As a practical application of BiSS, we show how to use its predictions to reduce the time required by another system to learn strong heuristic functions from days to minutes in the domains tested.