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 ff be the number of arctic foxes. Then 4f4f is the number of their legs. Similarly, let ss be the number of snowy owls and 2s2s be number of their legs, Then,

4f+2s=84 and 84>2(f+s),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 and 42>f+s.2f + s = 42 \text{ and } 42 > f+s.

Suppose for a moment there are no owls: ss = 0. Then there are 21 foxes and both equation and inequality are satisfied. If there are snowy owls, that is, s>0,s >0, then by the equation there must be an even number of them, since two owls make one fox (in legs). Thus ss might be 0,2,4,0, 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=0f = 42, s = 0

f=41,s=2f = 41, s = 2

f=40,s=4f = 40, s = 4

⋯\cdots

f=1,s=40f = 1, s = 40

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