Imagine we have a finite group $G$ of order $n$, with group elements ordered as $g_1,\dots,g_n$ in some unknown order. Now suppose we make random queries to its multiplication table, picking a uniform random $(i,j)\in[n]^2$ and determining the triple $(i,j,k)$ such that $g_ig_j=g_k$. How many queries are typically needed before we can reconstruct the entire multiplication table? What if we make online deterministic queries instead, i.e., we are allowed to specify $(i,j)$ in each query based on the results of previous queries? In the first case we show that for (most) abelian groups the first time at which we can reconstruct the multiplication table is, with high probability, the first time that we have encountered at least $n-1$ elements. This typically occurs at about $\frac{n}{3}\log n$ queries. For the second case, $(1+o(1))n$ is enough for many (even non-abelian) groups, and we can usually do slightly better than $n$ in the abelian case.
Joint work with Freddie Illingworth, Alex Scott, Maya Stein and Youri Tamitegama.