Decomposing Euclidean space with a small number of smooth sets

Let the cardinal invariant sn denote the least number of continuously smooth n-dimensional surfaces into which (n + 1)-dimensional Euclidean space can be decomposed. It will be shown to be consistent that sn is greater than s(n+1). These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.