Kangaroo Hops
Kanga hops ten times in one of four directions. At how many different points can he end up?
Problem
A kangaroo is sitting in the Australian outback. He plays a game in which he may only jump 1 metre at a time, either North, East, South or West. At how many different points could he end up after 10 jumps?
Student Solutions
Consider the kangaroo's starting position as the origin of coordinate axes, with East and North being the positive x and y directions, respectively, and one metre being one unit along the axes.
We begin by considering the first quadrant. If the kangaroo's end point has coordinates $(a,b)$, then a and b must be integers. Also, after $10$ jumps, it must be that $a+b \leq 10$. Hence his end points are bounded by the right-angled triangle with vertices at $(10,0)$, $(0,10)$ and $(0,0)$.
He can finish at any point on the hypotenuse of the triangle since all these points satisfy $a+b=10$ and so can be reached by a jumps East and b jumps North. But he can only end up at a point $(a,b)$ on the other two edges or inside the triangle if $a+b$ is even. (He can certainly reach all such points in $a+b\leq
10$ jumps and if $a+b$ is even, with $a+b< 10$, he can jump away and back again using up $2$ jumps, and can repeat this until he has made $10$ jumps, and so end up at $(a,b)$.)
By symmetry we see that the possible end points form a square of side $11$, and so there are $121$ of them, as shown in the diagram.
