Problem (Geometry):
A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it travels 3 units down to $(2,-2)$. On move 4 , it travels 4 units to $(-2,-2)$. It continues in this fashion, so that on move $n$, it turns $90^{\circ}$ in a clockwise direction from its previous heading and travels $n$ units in this new direction. After $n$ moves, the goat has travelled a total of 55 units. Determine the coordinates of its position at this time.

Solution:
After 2 moves, the goat has travelled $1+2=3$ units.

After 3 moves, the goat has travelled $1+2+3=6$ units.

Similarly, after $n$ moves, the goat has travelled a total of $1+2+3+\cdots+n$ units.

For what value of $n$ is $1+2+3+\cdots+n$ equal to 55 ?

The fastest way to determine the value of $n$ is by adding the first few integers until we obtain a sum of 55 . This will be $n=10$.

(We could also do this by remembering that $1+2+3+\cdots+n=\frac{1}{2} n(n+1)$ and solving for $n$ this way.)

So we must determine the coordinates of the goat after 10 moves.

We consider first the $x$-coordinate.

Since starting at $(0,0)$ the goat has moved 2 units in the positive $x$ direction, 4 units in the negative $x$ direction, 6 units in the positive $x$ direction, 8 units in the negative $x$ direction and 10 units in the positive $x$ direction, so its $x$ coordinate should be $2-4+6-8+10=6$. Similarly, its $y$-coordinate should be $1-3+5-7+9=5$.

Therefore, after having travelled a distance of 55 units, the goat is at the point $(6,5)$.

Final answer: (6,5)