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.)