The Mad Veterinarian

I volunteered at the Julia Robinson Math Festival held at a nearby school last Saturday. It was a great event for kids in 4th to 9th grade in the area, featuring a room full of mathematical activity tables and raffle tickets to be given out as rewards for a good idea or explanation.

The table I was working at featured a number of problems about a ``mad veterinarian’’. He had a machine that you can feed two cats to and get a dog out. You can also feed it two dogs to get one dog out, and if you feed it a cat and a dog, you get out a cat. The veterinarian wants to end up with only one cat, and no other animals. Can he do this starting with three cats and one dog? How about four cats and two dogs?

After playing around with this a bit, one realizes that a simple parity argument shows which starting positions he can get just one cat from. But here’s a harder question:

He built a new machine that now can only accept two different types of animals, and returns either a dog, a cat, or a mouse, as follows. If you feed it a cat and a dog, you get out a mouse. If you feed it a dog and a mouse, you get out a cat. If you feed it a mouse and a cat, you get out a dog. Now, for which starting numbers of cats, dogs, and mice can he end up with exactly one cat in the end (and no other animals)?

Needless to say, it was a fun morning of mathematics. :)