**Category:** Number Concepts, Patterns, Measurements, Variables & Equations, Transformations, Number Operations, and Relations & Functions**Suitable for Grade Level:** Elementary and Secondary

**The Math in this Problem:**

Vectors are the basis to this math puzzle, where students will apply their knowledge of directions and lengths onto a grid. With the acceleration of each consecutive vector, students are challenged to derive a solution for finding a way back to the coyote’s initial position.

Coyotes demarcate their territory by peeing on bushes, rocks, shrubs, cacti, trees, and anything else that is lying around. It is tiring, never-ending work. That is why one especially intelligent coyote decided to order rocket-propelled shoes to allow the more efficient demarcation of territory and therefore leave more time for leisure activities like reading Euclid and eating birds.

The problem was that, once started, the shoes continued to accelerate, but in a funny way. The first minute, the shoes traveled 1 km, then turned left or right as the coyote commanded. The second minute the shoes traveled 2 km, then turned left or right as the coyote commanded. The third minute the shoes traveled 3 km, then turned left or right as the coyote commanded…

The only thing that would stop the shoes from accelerating was if the coyote ended a minute at the exact spot where he turned the shoes on.

Example: The coyote after the sixth minute is only 3 km from his starting place…

Find a series of left and right turns that will allow the coyote to get back to his starting place in 10 minutes or less.

## Hint

In order for the Coyote to get back home, two things must be true. The Easterly movement must balance the Westerly movement and the Northerly movement must balance the Southerly movement.

## Further Hint

Assume that the Coyote starts by going North. (Convince yourself that if we solve the problem with this initial direction, then we can solve it for any initial direction.)

The North-South steps will always be odd integers. The East-West steps will always be even integers.

1) Let’s consider North-South first. Is it possible that after an odd number of odd steps, that the Coyote will be back home? For example, can you assign the first 11 odd numbers to either North or South so that the total distance travelled by North and South is the same?

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21

(If you are having trouble, ask yourself whether the distance travelled North can be even if the distance travelled South is even? Can the distance travelled North be odd if the distance travelled South is odd?)

2) Now let’s consider East-West. Is it possible to assign the first six even numbers to East and West so that the distances are equal?

2, 4, 6, 8, 10, 12

(If you are having trouble, ask if West’s total distance is divisible by 4, can East’s total distance be divisible by 4.)

Is this solution unique?

#### Extensions:

- The coyote must enclose a territory so the following sequence of turns is not acceptable:

- Whereas this one is acceptable:

- The coyote starts his stopwatch before he begins, and checks it again when he returns. How long can it take the coyote to get home? For example, can it take him exactly an hour, a day?