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Minnesota State University, Mankato
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Do dogs know calculus? It appears Elvis has the knack

Visiting professor Tim Pennings shows Minnesota State Mankato audience his dog’s knowledge of calculus.

Amanda Dyslin, Mankato Free Press, 3-21-2012

MANKATO — Years ago, Tim Pennings and his Welsh corgi, Elvis, were playing ball at the beach along Lake Michigan. Or, arguably, that’s what Elvis thought was going on.

Thirty-five times exactly, Pennings stood with Elvis at a set point on the beach and threw a ball at approximate­ly the same point in the water, and Elvis tore down the beach for a ways before leaping into the water like a champ to retrieve his prize.

What made this little game of fetch a bit unusual was the fact that Pennings, after throwing the ball, would tear down the beach after his dog with a screw­driver, which he would plunge into the sand at the exact point where Elvis chose to quit running on land and start swimming.

And for all 35 throws, he calculated the distance of Elvis’ run, the distance of his swim — and using a complicated equation requiring a solid back­ground in math — he determined whether Elvis himself had been using cal­culus.

Could a dog be gauging the shortest possible time it would take to get to the ball by knowing how fast he runs versus how fast he swims? Taking into account the distance to the ball from the shore at each moment?

Pennings, a math profes­sor at Hope College in Michigan, presented this theory Tuesday afternoon at a math colloquium, “Do Dogs Know Calculus?” at Minnesota State University.

It’s a talk he’s given at numerous colleges, on the BBC and National Public Radio, and on a number of talk shows. Even the “Ellen” show invited him on, although a scheduling conflict got in the way.

Elvis’ fame all stems back to the first trip to the beach with Pennings, after he adopted the dog. Pennings noticed that when he threw the ball at an angle to the right out into the water, Elvis didn’t immediately leap into the water and start swimming, nor did he run all the way down the beach and swim in a straight line to the ball. He chose to run a ways down the beach to a certain point and then swim a lesser dis­tance at an angle toward the ball.

“As soon as I saw him doing that something clicked in my mind,” Pennings said. “I wondered how close he was coming to jumping in at the optimal point. ... Elvis’ only objec­tive is to get to that ball the quickest.”

The 35 test throws proved it to Pennings: Within about a foot on most throws, Elvis was choosing the point on the beach to stop running and start swimming that result­ed in the least time for him to reach the ball. Almost every time.

“ That’s kind of neat,” Pennings said.

In front of 50 or so math students, Pennings spent 20 minutes or so diving into the formula for which he used to calculate Elvis’ amazing feat. While dis­cussing powers and deriva­tives and factors and solv­ing for y, Elvis laid on his side panting, tired from four runs down the hallway after a tennis ball. Any signs of recognition over what were supposedly his own calculations happening on the white board were nil. Pennings admits, when he asks Elvis calculus ques­tions, such as “What’s the derivative of X squared?” he doesn’t get much of a response.

“He just sits there,” Pennings said, which is actually, in a way, a much more interesting outcome.

“He just has this natural knack for finding the best solutions. ... Nature has the way of finding the best solu­tions.”

After Pennings wrote an article about his findings, a professor reading the piece suggested Elvis’ knack boils down to perception — that at each moment he’s run­ning down the beach, he’s assessing how close he is to the ball and simply jumping in when the ball begins to appear close enough to swim to quickly.

Of course, being a mathe­matician, Pennings couldn’t just assume. He had to find out.

Pennings got into the water with Elvis during their next visit to the beach and threw the ball to a point in the water parallel to the beach. Based solely on Elvis’ perception of the ball at his starting point already in the water, logical­ly, he would simply swim directly to the ball.

Instead, Elvis swam the short distance to shore, ran a good length of the beach, and swam back out to the ball, which was the route that got him to the ball the quickest.

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