Mathematics of pickleball

Trish Hammer, an alumna who is now associate dean for faculty affairs in the Virginia Tech College of Science, will compete in the USA Pickleball National Championship in Dallas in November. Unlike most of her opponents, Hammer is utilizing her mathematical training and analytical thinking skills to give her an edge.

With a bachelor's, master's, and doctorate in mathematics from Virginia Tech, Hammer’s perspective as an applied mathematician reframes the popular sport.

“You use [math] all the time,” Hammer said. “You don’t even know that you’re using it.”

Because Hammer is aware of the concepts of math in pickleball, she is upping her game.

“One big application of math in the calculus courses I teach is optimization, and for me, pickleball presents itself as one big optimization problem. I am trying to maximize output based on the variables I can control,” said Hammer. “Maximizing means winning and I’m always thinking about adjusting the variables to beat an opponent.”

Variables include techniques, shot selection, game strategies, conditioning, diet, and instruction. Hammer pulls from these resources which translates into one big optimization problem.

“You are constantly adjusting the different skills, strategies, and conditioning, and I approach it all, surprisingly, very analytically,” said Hammer.

In addition to analyzing the game through a lens of optimization, other areas of mathematics also come into play. Hammer uses angles and margins of error to determine shot placement.

“I’m trying to place a shot that optimizes my chances for success,” said Hammer. “With each of those shots, you have to think about the angle, like 45 degrees, 80 degrees, and 10 degrees, while also thinking about the margin of error.”

If your opponent is standing in the back of the court, hitting a tighter angle is usually a more effective shot but it also comes with a greater margin of error.  The goal is to find the “sweet spot” between angle and margin error.

“If I wanted to minimize my margin error, I would just hit it straight back to the opponent. Safe, but not very effective, so you’re always trying to weigh that balance,” said Hammer.

A pickleball player might be tempted to choose a hit farthest from an opponent, but Hammer said that may not be the hardest shot to return, yet another variable for optimization.

“Take into account how quickly your opponent moves in different directions,” said Hammer. “An opponent may move more slowly side to side, and if so, it may be more effective for me to hit it a little closer to them by their side which has a smaller margin of error.”

The same concept applies to reacting to an opponent’s shots.

“Should I move up and then over? Should I just take the shortest path?  What movement puts me in the best position to react? I think about it all the time, that the direct distance may not necessarily be the quickest distance,” said Hammer.

Permutations in pickleball is where the mathematical possibilities really expand according to your skill level. Take the dink shot for example. It’s a soft hit, a little like pingpong, where players hit the ball softly back and forth at the no-volley zone.

“The dink is a key to the game because you’re trying to move the opponent and you’re trying to get them to hit it high enough that you can smash it and win a point,” said Hammer.

Though it’s a very nuanced and delicate shot, there are lots of dink possibilities. At a very basic level, there are three places for a player to hit the dink: straight across the court, toward the center of the court, or diagonally across the court. A rally might involve three shots, which means with three choices for each shot, there would be 27 ways to sequence three shots.

However, an advanced player is also considering placement, speed, and spin.

“All of a sudden for every dink you hit, you have three placement choices, two speed choices, and three choices for spin. Every time I hit the ball, I have 18 different ways to hit this one shot which translates to 5,832 ways to sequence three shots. Imagine the possibilities if the sequence is 10 or more shots.  So this whole idea of permutations is a big deal. The idea is you want to consider these permutations in a way that optimizes your chances for success,” said Hammer.

Pickleball comes down to patterns, and mathematics describes those patterns.

“When you’re playing an opponent, you’re always looking for patterns,” said Hammer.

A fellow pickleballer told Hammer that when opponents get ready to hit a shot, he analyzes it this way: There is a 75 percent chance the shot will land in one place, a 20 percent chance it will land in a second location, and a 5 percent chance it will go in yet another direction. Based on that calculation, he prepares for the locations the shot is most likely to land.

“And I said, ‘Oh my gosh, I do the same thing, but I had never heard it explained quite that way.’ That’s exactly how you’re thinking when you play,” said Hammer.

Although Hammer’s friend and most players are not mathematicians, they all use mathematical calculations, in pickleball and in everyday life.

“People don’t think that there’s math around them, that they never use it, but we all do without realizing it,” said Hammer. “I’m aware of it and the more aware of it I can be, this becomes a strength of my game.”