Take the 2-minute tour ×  Consider a square planar grid. (The
vertices are pair of points in the plane with integer
coordinates and two vertices are adjacent if they agree in one
coordinate and differ by one in the other.)  Give every edge a
length one with probability a half and length two with
probability a half.  Consider a shortest path between the origin
and the vertex $(n,0)$.  Show that with probability that tends
to one as $n$ tends to infinity the shortest path will not
contain the "middle edge" on the x-axis inbetween the orgin and
$(n,0)$. (Namely, the edge between the vertices
$(\lfloor\frac{n}{2}\rfloor,0)$ and
$(\lfloor\frac{n}{2}\rfloor+1,0)$.)  This question is in the
category of "a missing lemma". It is not really a full fledged
open problem but rather a statement which looks correct that was
needed in some paper and resisted proof. Of course, some such
"lemmas" turn out to be very difficult, but sometimes maybe a
simple argument was simply missed. The relevant paper is with
Itai Benjamini an