Deciding the existence of an $l\times m\times n$ integer threeway table with given line-sums is NP-complete already for fixed $l=3$, but is in P with both $l,m$ fixed. Here we consider {\em huge} tables, where the variable dimension $n$ is encoded in {\em binary}. Combining recent results on integer cones and Graver bases, we show that if the number of {\em layer types} is fixed, then the problem is in P, whereas if it is variable, then the problem is in NP intersect coNP. Our treatment goes through the more general class of $n$-fold integer programming problems.