[Math Lair] Mathematics for Golfers

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It is only quite recently that I have taken up golf. In fact I have played it for only three or four years, and seldom more than ten games in a week, or at most four in a day. I have had a proper golf vest for only two years. I bought a "spoon" only this year and I am not going to get Scotch socks till next year.

In short, I am still a beginner. I have once, it is true, had the distinction "of making a hole in one," in other words, of hitting the ball into the pot, or can, or receptacle, in one shot. That is to say, after I had hit, a ball was found in the can, and my ball was not found. It is what we call circumstantial evidence—the same thing that people are hanged for.

Under such circumstances I should have little to teach to anybody about golf. But it has occurred to me that from a certain angle my opinions may be of value. I at least bring to bear on the game all the resources of a trained mind and all the equipment of a complete education.

In particular I may be able to help the ordinary golfer, or "goofer"—others prefer "gopher"—by showing him something of the application of mathematics to golf.

Many a player is perhaps needlessly discouraged by not being able to calculate properly the chances and probabilities of progress in the game. Take for example the simple problem of "going round in bogey." The ordinary average player, such as I am now becoming—something between a beginner and an expert—necessarily wonders to himself, "Shall I ever be able to go around in bogey; will the time ever come when I shall make not one hole in bogey, but all the holes?"

To this, according to my calculation, the answer is overwhelmingly "yes." The thing is a mere matter of time and patience.

Let me explain for the few people who never play golf (such as night watchmen, night clerks in hotels, night operators, and astronomers) that "bogey" is an imaginary player who does each hole in the fewest strokes that a first-class player with ordinary luck ought to need for that hole.

Now an ordinary player finds it quite usual to do one hole out of the nine "in bogey" as we golfers, or rather, "us goofers" call it; but he wonders whether it will ever be his fate to do all the nine holes of the course in bogey. To which we answer again with absolute assurance, he will.

The thing is a simple instance of what is called the mathematical theory of probability. If a player usually and generally makes one hole in bogey, or comes close to it, his chance of making any one particular hole in bogey is one in nine. Let us say, for easier calculations, that it is one in ten. When he makes it, his chance of doing the same with the next hole is also one in ten; therefore, taken from the start, his chance of making the two holes successively in bogey is one-tenth of a tenth chance. In other words, it is one in a hundred.

The reader sees already how encouraging the calculation is. Here is at last something definite about his progress. Let us carry it farther. His chance of making three holes in bogey one after the other will be one in 1000, his chance of four one in 10,000, and his chance of making the whole round in bogey will be exactly one in 1,000,000,000—that is one in a billion games.

In other words, all he has to do is to keep right on. But for how long? he asks. How long will it take, playing the ordinary number of games in a month, to play a billion? Will it take several years? Yes, it will.

An ordinary player plays about 100 games in a year, and will, therefore, play a billion games in exactly 10,000,000 years. That gives us precisely the time it will need for persons like the reader and myself to go round in bogey.

Even this calculation needs a little revision. We have to allow for the fact that in 10,000,000 years the shrinking of the earth's crust, the diminishing heat of the sun, and the general slackening down of the whole solar system, together with the passing of eclipses, comets, and showers of meteors, may put us off our game.

In fact, I doubt if we shall ever get around in bogey. Let us try something else. Here is a very interesting calculation in regard to "allowing for the wind."

I have noticed that a great many golf players of my own particular class are always preoccupied with the question of "allowing for the wind." My friend Amphibius Jones, for example, just before driving always murmurs something, as if in prayer, about "allowing for the wind." After driving he says with a sigh, "I didn't allow for the wind." In fact, all through my class there is a general feeling that our game is practically ruined by the wind. We ought really to play in the middle of the Desert of Sahara where there isn't any.

It occurred to me that it might be interesting to reduce to a formula the effect exercised by the resistance of the wind on a moving golf ball. For example, in our game of last Wednesday, Jones in his drive struck the ball with what, he assures me, was his full force, hitting it with absolute accuracy, as he himself admits, fair in the center, and he himself feeling, on his own assertion, absolutely fit, his eye being (a very necessary thing with Jones) absolutely "in," and he also having on his proper sweater—a further necessary condition of first-class play. Under all the favorable circumstances the ball advanced only 50 yards! It was evident at once that it was a simple matter of the wind: the wind, which was of that treacherous character that blows over the links unnoticed, had impinged full upon the ball, pressed it backward, and forced it to the earth.

Here, then, is a neat subject of calculation. Granted that Jones—as measured on a hitting machine the week the circus was here—can hit 2 tons, and that this whole force was pressed against a golf ball only one inch and a quarter in diameter. What happens? My reader will remember that the superficial area of a golf ball is πr³, that is 3.141567 × (5/8 inches)³. And all of this driven forward with the power of 4000 pounds to the inch!

In short, taking Jones's statements at their face value, the ball would have traveled, had it not been for the wind, no less than 6½ miles.

I give next a calculation of even more acute current interest. It is in regard to "moving the head." How often is an admirable stroke at golf spoiled by moving the head! It have seen members of our golf club sit silent and glum all evening, murmuring from time to time, "I moved my head." When Jones and I play together I often hit the ball sideways into the vegetable garden from which no ball returns (they have one of these on every links; it is a Scottish invention). And whenever I do so Jones always says, "You moved your head." In return when he drives his ball away up into the air and down again ten yards in front of him, I always retaliate by saying, "You moved your head, old man."

In short, if absolute immobility of the head could be achieved, the major problem of golf would be solved.

Let us put the theory mathematically. The head, poised on the neck, has a circumferential sweep or orbit of about 2 inches, not counting the rolling of the eyes. The circumferential sweep of a golf ball is based on a radius of 250 yards, or a circumference of about 1600 yards, which is very nearly equal to a mile. Inside this circumference is an area of 27,878,400 square feet, the whole of which is controlled by a tiny movement of the human neck. In other words, if a player were to wiggle his neck even 1/190 of an inch the amount of ground on which the ball might falsely alight would be half a million square feet. If at the same time he multiplies the effect by rolling his eyes, the ball might alight anywhere.

I feel certain that after reading this any sensible player will keep his head still.

A further calculation remains—and one perhaps of even greater practical interest than the ones above.

Everybody who plays golf is well aware that on some days he plays better than on others. Question—How often does a man really play his game?

I take the case of Amphibius Jones. There are certain days, when he is, as he admits himself, "put off his game" by not having on his proper golf vest. On other days the light puts him off his game; at other times the dark; so, too, the heat, or again the cold. He is often put off his game because he has been up late the night before; or similarly because he has been to bed too early the night before; the barking of a dog always put himself off his game; so do children; or adults; or women. Bad news disturbs his game; so does good; so also does the absence of news.

All of this may be expressed mathematically by a very simple application of the theory of permutations and probability; let us say that there are altogether 50 forms of disturbance any one of which puts Jones off his game. Each one of these disturbances happens, say, once in ten days. What chance is there that a day will come when not a single one of them occurs? The formula is a little complicated, but mathematicians will recognize the answer at once as
+ ........
. In fact, that is exactly how often Jones plays at his best;
worked out in time and reckoning four games to the week and allowing for leap years and solar eclipses, comes to about once in 2,930,000 years.

And from watching Jones play I think that this is about right.

Also by Stephen Leacock are Boarding-House Geometry, The Force of Statistics, and The Mathematics of the Lost Chord.

More mathematics humour.