2021 JMC 10 Problems/Problem 20
Problem
A particle is in a grid. Each second, it moves to an adjacent cell and when traveling from a cell to another cell, it takes one of the paths with shortest time. The particle starts at cell
and travels to cell
in
seconds, to cell
in
seconds, and finally back to cell
in
seconds. How many possible triples
exist?
Solution
We do cases on whether to
is three steps in one direction or two steps in one direction and one step in a perpendicular direction.
Case 1. Three steps straight from to
.
The red dots mark the places of squares that are 5 away from and the blue dots mark the squares that are 4 away from
. The two purple dots are valid placements for
However, we can also have
on top and
on bottom, or
and
aligned horizontally, so we have
symmetries. There are also
ways to embed the
hull of
into our
grid, so our total for this case is
.
Case 2. Two steps one direction, one step in a perpendicular direction from to
.
We can embed the hull in
ways (we only have a
grid), a
or
hull in
ways, and a
in
ways. This gives
ways. However, we must multiply this by
to accommodate for the
positions (symmetric) of
relative to
giving
cases here. Our final answer is
.