Elvis: the Welsh Corgi Who Knows Calculus

[Excerpted from Chapter Two of The Math Instinct]

“There’s something odd about the way Elvis runs to fetch the ball,” Tim Pennings thought to himself one day in 2001.

As he did several times a week, Pennings, of Holland, Michigan, had brought his Welsh Corgi dog Elvis down to the shore of Lake Michigan to play fetch.

Sometimes, Tim would throw the ball along the beach and watch as his dog streaked along the sand to retrieve it. Other times, he hurled the ball out into the water, and that was when he noticed Elvis’s strange behavior. If Tim threw the ball straight out into the water, Elvis rushed into the Lake and swam directly for it. But if he threw the ball into the water in a diagonal fashion, slanted toward the beach, then instead of simply heading off straight toward the ball, Elvis ran along the water’s edge for a while before diving in.

Thousands of dog owners must have seen exactly the same behavior and thought nothing of it. But Pennings is an Associate Professor of Mathematics at Hope College in Michigan, and Elvis’s behavior reminded him of a calculus problem he often gave his students to solve. Not just that, but as far as Tim could tell, Elvis was getting the right answer, which is more than he could say for many of his students. “Can my Welsh Corgi do calculus?” he wondered.

Tim knew the answer had to be no, but after throwing a diagonal ball into the water a few times and watching the path Elvis took to reach it, he was sure something very interesting was going on. What Elvis seemed to be doing was choosing a path that got him to the ball in the shortest time. But the only way Pennings knew to figure out that path was to use calculus.

Chasing a ball thrown along the sand or else straight out into the lake, the quickest way to reach it is by a straight line directly to the ball. But with a ball thrown out into the lake diagonally, it’s much more complicated. Because a dog can run much faster on land than it can swim in water, it is quicker first to run along the water’s edge some distance and then dive in and swim the remainder. One way to do this would be to run along the water’s edge until directly opposite the ball — by now bouncing on the surface of the water — and then make a sharp, right-angle turn into the lake and swim out toward it. But a much quicker way is to run along the beach part way toward the point opposite the bobbing ball, and then swim diagonally from that point in a straight line toward the ball. The question is, in order to get to the ball quickest, exactly how far along the beach should Elvis run before jumping into the water?

This is a classic problem that math professors regularly give to their students. The diagram shows how a college math student is supposed to solve it. The solution requires calculus, a deep mathematical technique discovered by the mathematicians Isaac Newton and Gottfried Leibniz in the 17th century.

The ball chasing problem. The dog starts at A and the ball is at B. The dog is supposed to run along the beach from A to a point D, and then dive into the water and head directly for B. The problem is to determine the length of the line AD for which the total time taken to reach B is least. To find the answer, you need to know the lengths AC and CB, the speed of the dog running on the sand, and the speed of the dog swimming.

Determined to understand just what Elvis was doing, Pennings set out to collect some data. On his next visit to the beach, he took along—in addition to the ball—a long tape measure, a stopwatch, and his swimming costume. Time after time—for a total of 35 repetitions in all—Tim hurled the ball, started the stopwatch, raced along the beach after his dog, dropped a marker at the point where Elvis dived into the water, noted the time it had taken Elvis to reach that point, and then followed him in, splaying the tape measure out behind him. Although Tim was left behind running on the beach, he is a strong swimmer, and was generally able to catch up with Elvis before the dog reached the ball, and was able to note the time it took Elvis to swim to the ball. Tim then swam back to the beach, noted the point where he hit land, and then used the tape measure to determine the actual lengths AD and AC. On average, Tim threw the ball 20 meters along the beach and 10 meters into the water. The entire process lasted three hours. Tim stopped when he was exhausted.

Once Tim got home with his measurements, some easy calculations allowed him to determine all the data required to solve the calculus problem — which he promptly did. As he had suspected, Tim discovered that, on average, Elvis took to the water at exactly the right spot given by calculus. The conclusion was inescapable: In his own way, Elvis was able to solve a college level calculus problem.

Tim wrote up his findings and published them in the May 2002 issue of The College Mathematics Journal, published by the Mathematical Association of America. The journal editor made it the lead article, under the title Do Dogs Know Calculus?, and put a photograph of Elvis on the front cover — most likely the first time ever that a dog has graced the cover of a mathematical journal.

So how was Elvis doing this? Here is how Pennings explained his findings:

... although he made good choices, Elvis does not know calculus. In fact, he has trouble differentiating even simple polynomials. More seriously, although he does not do the calculations, Elvis’s behavior is an example of the uncanny way in which nature (or Nature) often finds optimal solutions. ... (It could be a consequence of natural selection, which gives a slight but consequential advantage to those animals that exhibit better judgement.)

In other words, Pennings says, the mathematics behind Elvis’s remarkable behavior has been done by Mother Nature. By the process of evolution by natural selection, dogs have developed the ability to do by instinct—perhaps enhanced by experience—exactly what is required to reach the ball in the shortest possible time. In that sense, Elvis is able to solve that one particular calculus problem