  Mossel and Ross raised the question of when a random colouring of a graph can
be reconstructed from local information, namely the colourings (with
multiplicity) of balls of given radius. In this paper, we are concerned with
random $2$-colourings of the vertices of the $n$-dimensional hypercube, or
equivalently random Boolean functions. In the worst case, balls of diameter
$\Omega(n)$ are required to reconstruct. However, the situation for random
colourings is dramatically different: we show that almost every $2$-colouring
can be reconstructed from the multiset of colourings of balls of radius $2$.
Furthermore, we show that for $q \ge n^{2+\epsilon}$, almost every
$q$-colouring can be reconstructed from the multiset of colourings of
$1$-balls.
