An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Choose any three by three square of dates on a calendar page...
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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.
This problem is taken from the UKMT Mathematical Challenges.