## 2500.31 – Foxes and Owls

Siqiao and Aurora are observing species in a large and still-frozen region of arctic tundra. There are arctic foxes and snowy owls under observation. When asked to count them, the tricksy Siqiao reported that the number of legs, total, was 84---and that this was more than twice the number of heads. How many of each animal are there?

Solution

Let $f$ be the number of arctic foxes. Then $4f$ is the number of their legs. Similarly, let $s$ be the number of snowy owls and $2s$ be number of $their$ legs, Then

$4f + 2s = 84 \text{ and } 84 > 2(f+s),$

where the equation counts legs and the inequality states that 84 is more than twice the number of heads. Dividing by two gives simpler versions of equation and inequality:

$2f + s = 42 \text{ and } 42 > f+s.$

Suppose for a moment there are no owls: $s$ = 0. Then there are 21 foxes and both equation and inequality are satisfied. If there $are$ snowy owls, that is, $s >0,$ then by the equation there must be an even number of them, since two owls make one fox (in legs). Thus $s$ might be $0, 2, 4,$ etc. as long as the total number of animals is less than 42. So the complete solution includes these possible values:

$f = 42, s = 0$

$f = 41, s = 2$

$f = 40, s = 4$

$\cdots$

$f = 1, s = 40$

It appears that Siqiao, in her trickiness, didn’t give enough information. Aurora bops Siqiao on the head.