Hardy, Littlewood and Golf In the course of reading several papers about the mathematics of golf, I was surprised to …nd J.E. Littlewood cited as one of the …rst to solve a particular problem. This surprise turned to astonishment two days later when I found a reference to a completely di¤erent problem in a 1945 paper called "A Math- ematical Theorem about Golf" by G.H. Hardy. My reaction was due to the reputations of Hardy and Littlewood as pure mathematicians par excellence. G.H. Hardy (1877-1947) and J.E. Littlewood (1885-1977) are high on any serious list of the most important mathematicians of all time. Individually, they made deep and numerous contributions to analysis and number theory, publish- ing hundreds of papers each. Together, they formed what many consider to be the greatest mathematical collaboration ever. The Hardy-Littlewood conjec- tures in number theory contain important insights into the distribution of prime numbers. They revitalized the study of analysis in England, and the rigor that they brought to their work elevated all of British mathematics. A saying, at- tributed to Harald Bohr as a friend’s"joke" but expressing a common belief in the mathematics community, was that in the early 1900’sthere were only three great English mathematicians: Hardy, Littlewood and Hardy-Littlewood. Hardy and Littlewood were also well known for their collaboration with the fantastic Srinivasa Ramanujan, whose mysterious conjectures are still being investigated. After discovering the Hardy paper, I was concerned that I would next read that a page of one of Ramanujan’s notebooks contains a formula for the perfect golf swing. At that point, I feared, Rod Serling would materialize and explain how it was that I had become embedded ... in the twilight zone. Hardy was quite outspoken about the beauty and purity of mathematics. In his book A Mathematician’sApology, he wrote, "The mathematician’spatterns, like those of the painter’s or the poet’s, must be beautiful, the ideas, like the colours or the words, must …t together in a harmonious way. There is no per- manent place in the world for ugly mathematics." He also wrote, "I have never done anything ’useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least di¤erence to the amenity of the world." This statement is often quoted and, undoubtedly, often misunderstood. I believe that his primary intent was to a¢ rm the purity of his motives, that he worked on problems for the beauty of the mathematics and not for personal pro…t or an illusion of saving the world. Given this background, Hardy and Littlewood are extremely unlikely can- didates for contributors to the mathematics of golf. However, I did not know at …rst that both men followed cricket avidly. Hardy scoured the newspapers for cricket statistics and often characterized other mathematicians’abilities by comparing them to famous cricketers. In modern golf terms, he might call Isaac Newton "in the Tiger Woods class" and Brook Taylor "in the Rocco Mediate class." In addition, Hardy regularly played real tennis, the game from which our modern tennis is descended. In this light, perhaps an interest in golf is not shocking. For completeness, a brief description of Littlewood’s golf problem follows. 1 The balance of this article concerns Hardy’sgolf problem. Littlewood’sGolf Problem One chapter of Littlewood’s book A Mathematician’s Miscellany discusses problems from the Mathematical Tripos examination used to rank mathematics graduates of Cambridge University. Littlewood was himself a Senior Wrangler in 1905, this being the term for the candidate with the highest score on the Tripos. Hardy had been fourth wrangler in 1898. One of the problems involves a spinning sphere bumped from a perch on top of a horizontal cylinder. The student is to prove that the sphere follows the path of a helix as long as it remains in contact with the cylinder. In the …gure on the left, a ball sits on top of a cylinder. If it is pushed, it will start to roll down the cylinder. The …gure on the right shows a helix (spiral) superimposed onto the cylinder. Obviously, the ball would not stick to the outside of the cylinder and follow the helix all the way around. The challenge is to show that the sphere will follow the helix until it loses contact with the cylinder and falls o¤. Ball on a Cylinder Cylinder with Helix Littlewood then asked what would happen if the cylinder is turned upright and the ball rolls along the inside of the cylinder. This could represent a ball rolling around the edge of a golf cup. What is its path? The most reasonable guess is that the ball will spiral down to the bottom of the hole. You might argue about whether the path is a perfect helix or whether gravity stretches out 2 the helix. However, almost nobody guesses the actual path. Ball in Hole Helical Path? The mathematical assumptions are that gravity is the only force and that the ball stays in contact with the cylinder. This is an obvious situation for using cylindrical coordinates. In the two dimensions seen from above the hole, the motion is circular and is easily described in polar coordinates. The motion is periodic. The third dimension can be left as a spatial variable representing, for example, the height z of the ball above the bottom of the cup. Cylindrical coordinates The result is counterintuitive: the height z is also described by a periodic function. In a helix or stretched helix, the height z would be a strictly decreasing function. Because z is periodic, the ball does not roll to the bottom of the hole. It spirals down some, and then spirals back up! (And then back down and then back up and so on.) A full solution is given in "Golfer’sdilemma" by Gualtieri, 3 Tokieda, Advis-Gaete, Carry, Re¤et and Guthmann in the June 2006 American Journal of Physics. They show that the ratio of the period of the vertical motion to the period of the circular motion is 7=2. Mathematically, then, a ball entering the hole could roll down into the hole p and then roll up and back out! A possible path is shown below. A Mathematical Lip-Out This does not seem reasonable to most of us. The reality is that the assump- tion that the ball stays in contact with the cylinder is not often met on a golf course, especially since the metal cup does not extend all the way to the top of the hole. If the ball enters the hole near the edge rolling along the dirt portion of the hole, a collision with the metal cup will bounce it free. As well, a ball entering the hole is more likely to bounce o¤ the lip or drop down into the hole than it is to roll along the edge. Unless the ball is rolling, this mathematical analysis is not valid. However, any experienced golfer can tell you stories about putts that did go into the hole and came back out. The greatest golfer of our time, Tiger Woods, had a putt in the 2007 PGA Championship to shoot a 62, which would have been the lowest score ever shot in a major championship. The putt went most of the way in, did a quick 180 turn and came back out. The mathematics and physics of a rolling ball show that this can happen. In his Miscellany, Littlewood comments that, "Golfers are not so unlucky as they think." Somehow, I doubt that this makes Tiger Woods feels any better. Hardy’sGolf Problem The problem proposed by G.H. Hardy is motivated by a hypothetical match between two golfers of "equal" ability. If one golfer is much more consistent than the other, which golfer has the advantage? Another way to state the problem is 4 in terms of strategy, known to golfers as "course management." Golfers are often faced with a choice of attempting a risky shot or playing safely. For instance, they may choose to try to hit over a lake directly at the hole, or play safely around the lake but farther from the hole. Is it better to use a cautious strategy or a risky strategy? Hardy approached this question by imagining a golfer who can hit three types of shots: excellent (E), normal (N) or poor (P). The golfer plays a hole with a par (target score) of four. A player hitting four normal shots (NNNN) …nishes the hole with a score of 4. A poor shot adds one to the required number of shots. Therefore, a player who hits three normal shots and then a poor shot (NNNP) has not …nished the hole. Another normal shot (making the shot sequence NNNPN) …nishes the hole with a score of 5. However, another poor shot extends the hole to at least six shots.,The sequence NNNPPN …nishes the hole with a score of 6. By contrast, an excellent shot reduces the required number of shots by one. Thus, a shot sequence of NNE …nishes the hole with a score of 3. Poor shots and excellent shots are essentially inverses of each other. The sequence NPEN …nishes the hole with a score of 4. More examples follow. Shot Sequence Score NPNNN 5 PNNE 4 ENPPN 5 NEPPPN 6 ENN 3 Notice that a sequence can never end in P. A poor shot always adds one to the sequence. However, a sequence can end in E.