Secret of perfect pool

A usual day at a local bar's pool table inspired a Louisiana State University mathematician to come with a proof on the difficulty of certain shots in pool.

The incident took place when Rick Mabry was shooting pool with colleagues one midweek afternoon in the early 1990s. Suddenly an idea struck him and he jotted down his thoughts on a napkin.

But now his conclusions have appeared in the January issue of The College Mathematics Journal. Mabry was fascinated by what happens after your opponent accidentally sinks the white cue ball in 8-ball pool.

The rules are different across countries, but in most American bars you usually set the cue ball anywhere behind the "head string". This is a line, often unmarked, that runs across the width of the table one-quarter of the way down its length. 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 "straight-in" shot is clear - hit the cue ball dead-on and it hits the object ball dead-on, which drops in the pocket.

Mabry noticed that for an 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.

Somewhere between those extremes, he thought, must be the hardest possible set-up for the straight-in shot. But where?

First Mabry had to define "difficulty" in mathematical terms. The easiest shots are the most forgiving; ones where a player can still sink the ball despite making a large "shooter error" with the cue.

In difficult shots, even the smallest errant twitch sends the ball careening off course.

Mabry using trigonometry to devise a formula to describe the object ball's deviation from its intended destination (the pocket) as a function of shooter error and the distance between the cue ball and object ball.

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.

Now Mabry returned 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's 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 - touching and perfectly aligned with the pocket.

"I wanted everything to come out kind of the way I thought it should," New Scientist quoted Mabry, as saying.

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 "combination shot", 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's original error and make combo shots more difficult that 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.

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?

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.