  A colouring of the edges of an $n \times n$ grid is said to be
\emph{reconstructible} if the colouring is uniquely determined by the multiset
of its $n^2$ \emph{tiles}, where the tile corresponding to a vertex of the grid
specifies the colours of the edges incident to that vertex in some fixed order.
In 2015, Mossel and Ross asked the following question: if the edges of an $n
\times n$ grid are coloured independently and uniformly at random using
$q=q(n)$ different colours, then is the resulting colouring reconstructible
with high probability? From below, Mossel and Ross showed that such a colouring
is not reconstructible when $q = o(n^{2/3})$ and from above, Bordenave, Feige
and Mossel and Nenadov, Pfister and Steger independently showed, for any fixed
$\epsilon > 0$, that such a colouring is reconstructible when $q \ge
n^{1+\epsilon}$. Here, we improve on these results and prove the following:
there exist absolute constants $C, c > 0$ such that, as $n \to \infty$, the
probability that a random colouring as above is reconstructible tends to $1$ if
$q \ge Cn$ and to $0$ if $q \le cn$.
