I hit it perfectly, but it wouldn’t go in.
—Tom Kite
The image is iconic. Jack Nicklaus’s 12-foot putt to take the lead on the 71st hole of the 1986 Masters is one of the most replayed shots in golf history. Along with its critical role in Nicklaus’s record sixth win, this putt has been canonized because of a perfect camera angle and an inspired three-word call by announcer Verne Lundquist. Lundquist gave us a hopeful “maybe” as the ball crawled toward the hole and then, when it dropped, a triumphant “Yes, sir!” beautifully synchronized to Nicklaus’s celebratory two-arm vertical thrust.
Lundquist’s initial “maybe” shows that he knew how tricky this putt was. Nicklaus’s caddie and son, Jackie, initially read the putt to break to the right, but Jack thought that this would be offset by the tendency of putts at Augusta National to break toward Rae’s Creek, which was to the left. The putt did indeed start to move right before straightening out. However, Jack has said, “I’ve gone back and putted that putt again a hundred times … I’ve never found that ball to go left again. But it did that time.”1
Not everyone was as fortunate as Nicklaus that day. Playing in a later group, Tom Kite experienced the agony of defeat. His 18th hole putt to tie Nicklaus and force a playoff refused to break enough and slid agonizingly over the edge of the hole. Whether these events were luck or fate is not entirely a philosophical toss-up.2 The role of luck in putting is the topic of this chapter. As we will see, there is a surprisingly large element of chance in making a putt, even on the nicest greens in the world.
Here is a statistic that surprises almost everyone, including professional golfers. What percentage of 12-foot putts does a professional golfer make? If you haven’t already heard the answer, think about it for a minute and take your best guess. Data collected at PGA tournaments in 1988 showed that the pros made a mere 25.7% of their 12-foot putts, with data collected only on flat greens.3 This is an amazing statistic. The best golfers in the world missed almost three-fourths of their 12-foot putts! By 2009, the percentage of putts made from 12 feet had increased to about 32% on all greens. Percentages from other distances are given in chapter 7, as well as breakdowns of which pros are the most succesful from different distances.
Putting guru Dave Pelz found one cause for the pros’ struggles. He built a machine called the True Roller that reliably reproduced both line and speed on a putt.4 The True Roller is basically a ramp, carefully engineered so that the ball rolls smoothly down the ramp and onto the green without bouncing. Pelz can aim the ramp in whichever direction he wants, and the speed of the ball on the green is determined by how far up the ramp the ball starts. The True Roller was tested to verify that its putts did not vary in direction or speed.
Pelz set up the True Roller at Westchester Country Club on pro-am day of the PGA’s Westchester Classic tournament on a freshly cut green before play started. From 12 feet, the True Roller made 73% of its putts. Setting up on the same green after the day’s play, the True Roller could make only 30% of its 12-footers! The culprit here is the condition of the green after being trampled by numerous big guys with spikes. A closer examination of what is happening on the greens follows later. From the drastic deterioration of the Westchester greens over a day’s play, it is clear that green quality has a significant influence on the percentage of putts made. To be fair, the dramatic drop in green quality occurred on pro-am day, where there is a large volume of play with distracted amateurs, and it was in the era of destructive metal spikes on golf shoes.5
If the perfect stroke only produces 30% successes from 12 feet, we cannot get too critical about the pros making only 26% in 1988. To be consistent, though, how much credit can we give Jack Nicklaus for his pressure putt in the 1986 Masters? Maybe, as Nicklaus implicitly acknowledged, he had a little luck.
Engineers Frank Werner and Richard Greig found similar results with their own putting machine. Over a variety of distances, they determined that the standard deviation of the lateral position of the ball is about 2% of the distance of the putt.6 For our 12-foot putt, 2% is almost 3 inches. Werner and Greig’s findings imply that 32% of putts that start out perfectly on-line will be 3 inches or more off-line by the time they reach the hole. Since the radius of the hole is only 2.125 inches, all of those putts will miss. Of the putts that roll exactly 12 feet, almost half will be wide of the hole.
Figure 3.1 Outcomes of a perfectly struck 12-foot putt, with a standard deviation of 2% of distance
Figure 3.1 shows the likelihoods of different left-right positions of a perfectly struck 12-foot putt using the 2% standard deviation rule. Only 52% of the putts stay within the hole, while 26% go up to an inch-and-a-half wide, and another 14% finish up to 3 inches wide of the hole. If you are keeping score, that leaves 8% of the perfectly struck putts more than 3 inches wide of the hole! The news gets worse as the distance increases. From 25 feet, the standard deviation is up to 6 inches, so that about 5% of perfectly struck putts will be more than a foot wide of the hole. Approximately three-fourths of putts struck with the perfect line will stray outside the hole.
Finally, there is one more piece of bad news. Werner and Greig found that distance has the same 2% rule. That is, a putt that should roll exactly 12 feet will actually have a random distance with standard deviation equal to 2% of 12 feet.7 This is the same standard deviation (about 3 inches) as for the left-right movement of the ball, so that figure 3.1 gives the percentages for a putt struck from left to right with “perfect” speed (that is, a speed that should produce exactly 12 feet of roll). Ignoring the sideways movement of these putts, 24% of these putts stop short of the hole. Another 24% would go past the hole, although, if the line is accurate enough, these would go in.
Of course, the good news in all of this is that you can stand (or sit) tall at the 19th hole. That blown 6-footer on the 18th hole which you swear broke to the left? Just quote the above research: it might not have been your fault. There is a lot of randomness in putting, and putts get bumped significant amounts by small imperfections in the greens.
What should you do about this when you’re standing over that 6-footer on the 18th? For starters, relax. Shaking uncontrollably and screaming, “I have no chance. It’s all random!” will not help. And, if you are in the group in front of me, please do not try to “read” the footprints. That attention to detail is not attractive in a Ryder Cup competitor, much less a weekend golfer.
The important lesson to take from this is that the perfect speed on a putt is not the speed that allows the ball to die at the hole. As noted above, distance is subject to the random bumps of the green, and your perfect putt could stop inches short of the hole. In other words, “never up, never in” is good practical advice.8 Of course, it is possible to hit putts too hard, and a firm putt needs to hit the center of the cup. You do not get partial credit for lip-outs.
What is the best speed to hit a putt? Dave Pelz experimented with his True Roller and measured the speed at which the most putts actually went in the hole. In Pelz’s tests, a target distance of approximately 17 inches past the hole maximized the likelihood that the putt went in.9 This does not depend on the length of the putt. Unfortunately, if your target is 17 inches past the hole, sometimes the ball will speed well beyond the hole. If your main desire is to have a simple tap-in for your next putt, a more conservative approach would serve you well. However, in match play, facing a putt to halve the hole, Pelz says that 17 inches past the hole should be your target.
In stroke play, you might not want to be so bold. Instead of maximizing the probability of making the putt, you are probably more interested in minimizing the total number of putts taken. This is the approach of Werner and Greig.10 Being bold might result in more one-putts, but it is not a good play if it also results in many more three-putts. With this more conservative philosophy, the ideal distance does depend on the length of the putt as well as on the speed of the green and the quality of the putter. Recommended distances range from 5 to 15 inches past the hole for most situations, reaching 17 inches only on fast greens (12 on the Stimp meter11) and for narrow distance ranges (around 12 feet for a mediocre putter and 20 feet for a good putter).
One unusual aspect of Werner and Greig’s work is that, for a long putt by a high handicapper, the ideal target distance is short of the hole. This certainly contradicts the “never up, never in” motto. However, there is good mathematics behind it. For a weak putter at 50 feet from the hole, the only realistic goal is to lag the first putt close enough to escape with a two-putt. Of course, getting a perfect line is not likely. This presents us with a geometry question. For an off-line putt such as that in figure 3.2, at which point is the ball closest to the hole?
The governing principle is that, at the closest point, the line from the ball to the hole makes a right angle with the ball path. As shown in figure 3.3, this occurs at a point that is slightly short of the hole. This shows that, for a long putt by a mediocre putter, short can be a good play. None of this analysis takes into account the slope of the green. In many cases, you would rather leave yourself a 4-foot uphill putt than a 3-foot sidehill putt. For most putts, however, the best advice is to be confident, aim a few inches past the hole, and do not be surprised if the putt goes a little random on you.
Figure 3.2 Off-line putt: where is the ball closest to the hole?
Figure 3.3 Closest point, at right angles with the hole
Sadly, the bad luck does not necessarily end when the ball reaches the hole. Lip-outs are the next topic.
One of the most famous hustles in golf history may have ended with an obscure result from mathematical physics. There are several versions of the story, but Raymond Floyd’s goes like this. In 1965, Floyd was well established on tour with two victories. He was living in Dallas, Texas, and ventured out to Tenison Park Golf Course to check out its famous gambling scene. After winning a few bets, Floyd was approached by the legendary hustler Titanic Thompson, who had a big money venture in mind.
Thompson flew Floyd flew out to El Paso to challenge a local hotshot. When Floyd drove out to the course to practice, a young man grabbed his bag and got him set to play. Floyd introduced himself, said he was there to play some guy named Lee Trevino, and asked if the bag man knew him. “That’s me,” said Trevino. Several thousand dollars, a lot of money in 1965, was wagered on the match. Floyd strutted out and shot 65 … and lost by two. He was not amused to lose to an unknown who “hit this little old screamer, never gets the ball in the air.” So, he went out the next day and shot 64 … and lost again.12 Apparently, Thompson desperately needed money and backed Floyd for another round.13 After all, Floyd had just won the St. Paul Open and recently finished sixth in the U.S. Open. On the 18th hole, both players had eagle putts to shoot 63. Floyd drained his. In Floyd’s words, “I can still see Lee’s putt. There’s no way it should not have gone in the hole. His ball kind of dives down in the hole, then comes out and sits on the lip.” Counting himself lucky and not yet broke, Floyd left town and returned to the “easy” life on the PGA Tour.14
What does mathematics have to do with this story? It turns out that mathematicians have analyzed the path of a golf ball after it goes in the hole. In fact, a version of it was used as a question at Cambridge University in the famous Mathematical Tripos examination and has been commented on by J. E. Littlewood, one of the most famous mathematicians of all time.15 The assumption is that the ball rolls along the side of a cylinder (the cup) under the force of gravity. The rolling assumption is unlikely to be met in golf, since many putts bounce off the lip or drop directly to the bottom of the hole. However, a putt trying to enter the edge of the hole will sometimes roll, and then our assumption is met.
Most people expect that the ball would spiral down to the bottom of the hole. This is partly correct. Viewed from above, the ball does appear to move around its circular boundary at a constant rate. The surprising part of the solution is that the vertical motion of the ball turns out to be alternating periods of downward motion and upward motion.
To solve this problem, two mathematical techniques are used to transform a seemingly impossible problem into a difficult but workable one. The geometry of the ball’s path is three-dimensional, so three variables are needed to describe its position. One variable measures the vertical (height) position. The other two variables measure its position as seen from directly above the hole. For these two, the polar coordinates introduced in the previous chapter simplify calculations considerably (see figure 3.4). An obvious advantage of polar coordinates is that, for a ball rolling on the cylinder, the value of r is constant.
The other trick is to use a moving reference frame. This means that instead of choosing a particular point as the origin, as indicated in figure 3.4, the origin moves with the ball. Depending on the problem to be solved, it can be more convenient to place the origin at the center of the ball or at the contact point between ball and cylinder.16 The result is counterintuitive. Both the angle θ and the height z oscillate. This means that the ball alternately spirals down and then spirals up, then down and up again until it loses contact with the side. One of the ways for the ball to lose contact with the side is to spiral back up out of the hole. Figure 3.5 shows a possible path for a ball going down into the cylinder and then coming back out of the hole. Now that’s a power lip-out!
Finally, let’s take a more detailed look at the mathematics of bad luck on the green. In mathematics, modeling refers to the process of approximating some aspect of reality with a mathematical description. The construction of a good model depends on in-depth knowledge of the process being described mixed with reasonable guesses and simplifications. Like a scientific theory, a model is used to make predictions. If the predictions turn out to be highly inaccurate, then the model must be abandoned. Good predictions count mostly as circumstantial evidence of the model’s usefulness, not as proof that the model is correct.
Figure 3.5 A lip-out, after the ball dips in to the hole
We will look at two models of random bumps on the greens. When I have mentioned the variability of greens to colleagues, their speculations have included spike marks, loose dirt, the dimples on the ball, and the wrath of the gods (this last one, obviously, from one who plays a lot of golf). Dave Pelz has written extensively about footprints on the green being to blame.17 Some modeling will help us evaluate these ideas.
Model 1 is based on the idea that the ball is constantly buffeted by spike marks, dirt, stiff blades of grass, etc. Mathematicians refer to this as a “random walk” (also called a “drunkard’s walk,” which would give a different meaning to the reference to golf being “a good walk spoiled”18). Imagine someone who is generally walking straight ahead but whose every step tips slightly to the left or to the right, with left or right being equally likely on each step. The actual path that the person walks might look like figure 3.6, where the person’s first step leans to the left, the second step tips back to the right, the next two steps lean to the left, and so on.
We can model a putt under constant bombardment as a random walk with infinitesimally small step size. That is, take the path in figure 3.6, extend it for a very large number of steps outside the box, and then shrink it back to fit in the box. Repeat this process several times. You now have a path with so many bends that it is impossible to distinguish individual steps. An example of such a path is shown in figure 3.7.
Properly scaled, this is considered a very good model for Brownian motion, the random movement of tiny particles suspended in a fluid. The math in this model is quite elegant and historically important, as Albert Einstein himself did some of the early work in the analysis of Brownian motion.19 For our purposes, it is also verifiably wrong. The standard deviation of the lateral displacement of a Brownian motion path is proportional to the square root of distance. Werner and Greig found that the standard deviation of golf putts is proportional to the distance of the putt (the 2% rule mentioned earlier). Going from a 5-foot putt to a 20-foot putt, the standard deviation of golf putts increases by a factor of 4. The Brownian motion model has a standard deviation that increases by a factor of 
So, model 1 fails.
Figure 3.6 First six steps of a random walk from left to right: the first step tips to the walker’s left, the second back to the right, the next two to the left, and so on.
Model 2 is based on the idea that the ball must travel through footprints on the way to the hole. These indentations change the angle at which the ball moves. (By contrast, the ball in the random walk model is always aimed straight ahead, with random left/right bumps added.) To help define the variables used, figure 3.8 shows a putt of d feet that has been bumped x feet off-line and is moving at an angle A to the straight path.
We next look at the ball after it has moved forward a distance y feet along the angle A. At this point, it may or may not hit a footprint and deflect. To be precise, with some probability p its angle is altered, and with probability 1 − p its angle remains equal to A. If it does hit a footprint, then half the time the angle increases to A + b, and half the time the angle decreases to A − b. This is illustrated in figure 3.9.
We continue this process until the ball has traveled the full distance d and then measure the amount x that the ball is off-line. The angular deflection b can be chosen so that the standard deviation of x is two percent of d.20 This matches both the experimental data of Werner and Grieg and the physical mechanism described by Pelz. Figure 3.10 shows the path of a putt generated by model 2. While the path may look smooth and resembles a putt that breaks to the right, it is actually produced by model 2, a putt on a flat green which is struck straight at the hole and then deflected by a series of footprints.
Figure 3.8 The ball is x feet off-line, moving at an angle of A from the straight path
Figure 3.9 After moving y feet along angle A, the ball’s angle may change from A to A + b or A − b
Figure 3.10 A putt deflected by footprints
Model 2 matches one aspect of the experimental data, so it remains a possible explanation of how the irregularities of a green affect a putt. This model has the advantage of supporting Dave Pelz’s explanation that footprints leave the green a “lumpy” carpet that alters the trajectory of the putt as it travels toward the hole.21
To close the chapter, I want to fill in some of the details of model 2 above. If the mathematical “lie” becomes unplayable, you can safely pick up and “play through” to the next chapter.
As illustrated in figure 3.9, x measures how far off-line the ball has drifted. If the ball moves a horizontal distance of y at an angle of A, the change in x is y tan A. The total deflection x is thus
x = y tan A1 + y tan A2 + … y tan An,
where A1 is the angle for the first step, A2 is the angle for the second step, and so on for all n steps, assuming that each step is of equal length y. The angles can change by ±b at each step, but they do not have to change at all. In mathematics, a surprising amount of progress can be made just by giving names to quantities. I will denote the changes in the angle by c1, c2, and so on. If the putt starts out on-line, then the initial angle is 0 and A1 = c1, A2 = A1 +c2 = c1 +c2, and so on.
For “small” angles, tan A ≈ A. This approximation makes the remainder of the analysis much easier to follow (but is not needed to get a result). Then
We are interested in the standard deviation of x. The mean value of x is 0, which makes the standard deviation equal to 
, where E(x2) is the expected value or mean of x2. The randomness in our final formula for x resides in the changes c of the angle. I assume that the c’s are independent but not identically distributed. Each c is 0 with probability 1 − p and ±b with equal probabilities 
. However, the probability p is not constant. Thus, the c’s have mean 0 but different variances.
In particular, I assume that the probability of a deflection depends on the distance from the hole. The closer to the hole the ball is, the higher the probability of a deflection (the larger p is). The logic behind this assumption is that footprints will tend to pile up near the hole and be relatively sparse far from the hole. For a putt of any distance, assume that the p-value for the last y feet is some value q. This means that the probability pn associated with cn equals q. Assuming that the probability is inversely proportional to distance from the hole, the probabilty associated with cn−1 is 
, the probability associated with cn−2 is 
, so on. This means that 
, and so on. In terms of x, we get
The total distance of the putt equals yn, so the standard deviation of x is given by
which equals 2% of the distance if 
. The amount of deflection b is therefore connected to the probability q of a deflection near the hole. We can think of the parameters b and q as measures of the quality of the green.
