We present Sudokube, a novel system that supports interactive speed querying on high-dimensional data using partially materialized data cubes. Given a storage budget, it judiciously chooses what projections to precompute and materialize during cube construction time. Then, at query time, it uses whatever information is available from the materialized projections and extrapolates missing information to approximate query results. Thus, Sudokube avoids costly projections at query time while also avoiding the astronomical compute and storage requirements needed for fully materialized high-dimensional data cubes. In this paper, we show the capabilities of the Sudokube system and how it approximates query results using different techniques and materialization strategies.