FUD鈥檚 bar and grill in Shreveport, Louisiana, doesn鈥檛 look like the kind of place where mathematical proofs are born. The jukebox is good, the beer is cheap and a smoky haze lingers over the pool tables where mathematicians from nearby Louisiana State University unwind after a hard day at the blackboard.
Rick Mabry recalls shooting pool with colleagues one midweek afternoon in the early 1990s when inspiration struck. So he did what any mathematician would do. He reached for a napkin and jotted down his thoughts. As he left that afternoon, Mabry tacked his napkin to the wall of the bar. It didn鈥檛 linger there long 鈥� one can easily imagine one of Fud鈥檚 patrons, haunted by school algebra, ripping it down.
No matter. His inspiration eventually led to a proof on the difficulty of certain shots in pool, (p 49). Mabry admits that his results will come as no surprise to professional pool players. However, they might just give you the edge when you next hit the green felt, or want to hustle your friends with a spectacular-but-simple trick shot.
Advertisement
鈥淗ustle your friends with a spectacular trick shot: a combination shot with 39 balls is unmissable鈥�
Mabry was fascinated by what happens after your opponent accidentally sinks the white cue ball in 8-ball pool. The rules vary from country to country, but in most American bars, at least, you then get to set the cue ball anywhere behind the 鈥渉ead string鈥�. This is a line, often unmarked, that runs across the width of the table one-quarter of the way down its length.
Sharp shooter
Typical players tend to place the cue ball so that it makes a direct line with an object ball and a pocket. The appeal of this 鈥渟traight-in鈥� shot is clear: hit the cue ball dead-on and it hits the object ball dead-on, which drops in the pocket. A no-brainer of a shot.
Mabry noticed that for your average player, the shot seems easiest in one of two situations, when the cue ball and object ball are close together. Or when the object ball is close to the pocket and therefore far from the cue ball (see diagram). Somewhere between those extremes, he reasoned, must be the hardest possible set-up for the straight-in shot. The question is: where?
First Mabry had to define 鈥渄ifficulty鈥� in mathematical terms. The easiest shots are the most forgiving; ones where a player can still sink the ball despite making a large 鈥渟hooter error鈥� with the cue. In difficult shots, even the smallest errant twitch sends the ball careening off course.
Using trigonometry, Mabry devised a formula to describe the object ball鈥檚 deviation from its intended destination (the pocket) as a function of shooter error and the distance between the cue ball and object ball (see diagram). In that equation, the shooter error is represented as the angle between two lines: the direction the cue ball was supposed to travel in, and where it actually went. Small angles indicate accurate shooters, whereas large ones mean the shooter needs more practice, or fewer beers.
With this definition of difficulty in hand, Mabry could return to his original question: where is the hardest straight-in shot? He first considered the simplest case of a player who is so precise that Mabry could effectively ignore shooter error. Now the difficulty of the shot boils down to the distance between the cue ball and the object ball.
To find the answer, all Mabry needed to do was plot the difficulty function against distance. With shooter error out of the picture, the function is a quadratic equation, mathematical-speak for saying its graph is rainbow-shaped. Rather than pots of gold, the two ends represent the easiest scenarios, where the object ball is either adjacent to the cue ball or teetering over the pocket. And the graph鈥檚 highest point reveals the hardest shot. It showed that the hardest shot was where the object ball sat in the middle between the cue ball and the pocket. It also showed that the easiest shot of all is when the cue ball and the object ball are closest together 鈥� touching and perfectly aligned with the pocket.
To seasoned players, these findings are common sense. To Mabry, they were reassuring. 鈥淚 wanted everything to come out kind of the way I thought it should,鈥� he says. But that satisfaction was tinged with regret. He had been secretly hoping for a more exotic result.
Determined to find something interesting, Mabry plugged his equations into a computer program to model a range of scenarios.
His efforts paid off with a succession of cocktail-party-worthy factoids. He extended his original problem to a straight-in 鈥渃ombination shot鈥�, or combo, where the aim is to shoot the cue ball into an object ball, which then strikes a second ball, which in turns drops into the pocket.
Mabry found that the difficulty of the combo shot depends on the separation of the object balls. Large distances amplify the shooter鈥檚 original error and make combo shots more difficult than single ball shots.
His calculations show that the hardest 2-ball combo is five times as hard as the hardest single-ball shot of the same length. And when Mabry added more balls into the mix, he found the hardest-possible straight-in combos involve 7 and 8 balls, all lined up and evenly spaced between the cue ball and the pocket.
More surprising, Mabry found that combos become easier with 9 balls, and more, because the separation is smaller. So much so, that a shot with 15 balls is even easier than a 1-ball shot, provided you can hit the cue ball hard enough (see diagram). And Mabry predicts that a combo with 39 balls is unmissable because the balls are touching and have no room to deviate.
After wandering through these other scenarios, Mabry tackled one final question: what is the hardest shot for a lousy player? Earlier he had assumed that his players were pretty accurate, but he wanted to know what would happen if he let his angle of error get larger 鈥� though not large enough to miss the object ball altogether.
As the angle of error grew, that 鈥渞ainbow of difficulty鈥� began to change shape and the maximum shifted its position. In fact, when he crunched the numbers, Mabry turned up an eerily familiar figure: 1.6180339887鈥� To those in the know, and indeed to those who aren鈥檛, this number represents the 鈥溾��, the irrational number (1+鈭�5)/2. 1:1.618 is celebrated by artists for its pleasing aesthetics and it frequently shows up in nature 鈥� from the spiral of a conch shell to the number of petals on a flower. You can even find it in your pocket, in the proportions of your bank cards.
Mabry found the same mysterious, underlying number on a pool table. The hardest possible shot for a lousy player occurs when the distance between the cue ball and the object ball is precisely 1.618 times longer than between the object ball and the pocket. Why this should be the case isn鈥檛 clear 鈥� Mabry chalks it up to one of those things.
David Alciatore, a mechanical engineer at Colorado State University in Fort Collins and author of The Illustrated Principles of Pool and Billiards, gives Mabry鈥檚 equations a thumbs-up for efficiency and elegance. 鈥淚t鈥檚 a case where the simple geometry led to a pretty interesting result,鈥� he says.
As for Mabry, he has moved onto other problems and no longer visits Fud鈥檚. 鈥淚 don鈥檛 shoot pool anymore,鈥� he confesses. Back in the day, he recalls, 鈥淚 was a pretty decent player after a beer. But not after two.鈥�
