Problem 570

The grasshopper begins at the origin, $(0,0)$.

At $(0,0)$, the sum of the $x$ and $y$-coordinates is $x+y=0+0=0$.

Each jump (up, down, left or right) increases or decreases either the
$x$-coordinate or the $y$-coordinate by $1$, and thus increases or decreases $x+y$ by $1$.

Thus each jump changes the parity of $x+y$ (from even to odd or from odd to
even).

Since $x+y$ is even at $(0,0)$, then after the grasshopper jumps
exactly $6$ times, $x+y$ is even at its final location $(x,y)$.

We begin by determining at which points $(x,y)$ the grasshopper can finish, where
$x\geq0$ and $y\geq0$ (in the first quadrant or on the
axes bordering the first quadrant).

Since the grasshopper jumps $6$
times, then $x+y$ is at most $6$.

For $x+y=6$, the grasshopper can, for
example, jump $6$ units right to end
at $(6,0)$, or jump $5$ units right and up $1$ to end at $(5,1)$. Continuing in this way with $x+y=6$, the remaining possible ending
points are $(4,2)$, $(3,3)$, $(2,4)$, $(1,5)$, and $(0,6)$.

At the grasshopper’s ending point $(x,y)$, can $x+y=4$? (Recall that $x+y$ is even so we may ignore $x+y=5$, for example.) The grasshopper
could, for example, jump $1$ unit
right, then $1$ unit left, returning
to $(0,0)$ with $4$ jumps remaining. Thus for $x+y=4$, the possible ending points are
$(4,0)$, $(3,1)$, $(2,2)$, $(1,3)$, and $(0,4)$.

When $x+y=2$, the possible ending
points are $(2,0)$, $(1,1)$, and $(0,2)$.

And finally, the grasshopper could jump right $3$ times and left $3$ times to end at $(0,0)$.

To summarize the case for which $x\geq0$ and $y\geq0$, the grasshopper can end at

$9$ points with $x>0$ and $y>0$: $(5,1)$, $(4,2)$, ($3,3)$, $(2,4)$, $(1,5)$, $(3,1)$, $(2,2)$, $(1,3)$, $(1,1)$,
$6$ points on the positive
$x$-axis or positive $y$-axis: $(2,0)$, $(4,0)$, $(6,0)$, $(0,2)$, $(0,4)$, $(0,6)$,
the point $(0,0)$.

By symmetry, the grasshopper can end at the $9$ points that correspond to (i) in each
of the other $3$ quadrants for a
total of $9\times4=36$ points.

By symmetry, the grasshopper can end at the $6$ points that correspond to (ii) on the
negative $x$-axis or the negative
$y$-axis for a total of $6\times2=12$ points.

Thus, $N=36+12+1=49$.