Par Combinatorics

How many possible golf courses are there?

Well, it's a stupid question, of course. There are an infinite number of golf courses. Everything's a continuous variable on a golf course, from the length of the holes to the width of the fairway in the landing area to the depths of the bunkers, and there are an infinite number of potential factors in any event. But there is one factor that has a certain primacy (see the previous post for that) and that is distinctly discrete: par. So let me say that for the purposes of this article, I am defining an equivalence relation on the set of 18-hole golf courses where two courses are equivalent if each of their holes of a certain number has the same par. And now let me rephrase my question: under standard assumptions about the par composition of golf courses, how many different "equivalence classes" of golf courses are there? This is a question we can answer (though why we'd want to is a different story; predominantly if we are bored...). The basic answer is, a lot.


I have never seen a golf course with a par other than 70, 71, 72, or 73, and indeed only one par-73 that I can think of. So in any event, let's define some conventions and terminologies. I will call a golf course "standard" if it has 4 par-3 holes and a sufficient number of par-5s to make it reach the desired total par. Moreover, since it is highly unusual for a par-3 to be the opening or closing hole of a nine, I will call a course "superstandard" if it is standard and if none of the 1st, 9th, 10th, or 18th holes is par-3. Then, just out of personal interest, I will also look at symmetric courses, which are obviously courses where if the number of two holes adds up to 19 their pars are the same. Here's a shorthand notation: SSS72 means superstandard symmetric par-72. Since all three attributes are abbreviated "S," here's what I'll use to denote, say, a "super, symmetric" course that isn't standard: S-S70, for instance. (Note that a symmetric course can't be odd par.) Likewise --S means symmetric but not standard or super; S-- means super but neither standard nor symmetric; -S- means standard but nothing else. Okay, here goes.

-S-72: 3,063,060 (18 choose 4 times 14 choose 4, because you put 4 par-3s onto 18 holes and then have 14 spots left for the 4 par-5s.)
-S-71: 1,113,840
-S-70: 278,460
-S-73: 6,126,120

Thus, there are 10,581,480 possible "standard" golf courses with 4 par-3s and par 70, 71, 72 or 73.

SS-72: 1,002,001
SS-71: 364,364
SS-70: 91,091
SS-73: 2,004,002
-SS72: 756 (it's only 9 choose 2 times 7 choose 2, because one nine determines the other nine.)
-SS70: 252
SSS72: 441
SSS70: 147

St. Andrews is a course with just 2 3's and 2 5's, and it's also super and symmetric. There are just 56 possible courses meeting these criteria: seven choices to put the par-3 in on the front nine, and then eight choices for the front nine's par-5 once you've placed the par-3. 2/14/2 courses as a whole have 18,360 possibilities, while 3/13/2 courses would have 85,680 possibilities.

Some courses have 5 5s and 5 3s. There are 11,027,016 possible courses like this. We could also imagine a 5/9/4 par-71 arrangement or a 5/10/3 par-70, of which there are 6,126,120 and 2,450,448. Note that the 5/9/4 courses have the same number as the 4/9/5 par-73 courses, though I calculated them in a way that made this not obvious from the combinatorics. But that is, of course, as it should be. Add these to the earlier 10,581,480 unique standard courses and we get a total of 30,289,104 possible golf courses up to pars. If we require them to be super, i.e. no par-3s on the bookend holes, then it's 2,576,574.

Of course, as my not-an-expert-in-golf roommate points out, strictly speaking there are 3^18 different 18-hole golf courses with just par-3, par-4, or par-5 holes. That's 387,420,489. I find it interesting that reasonably normal golf courses are almost 10% of all possible golf courses: given that we're constraining ourselves strictly by having no more than 5 and no less than 2 of either kind of non-par-4, I think there must be a lot of courses in this zone relative to other, similarly defined zones.

I thought of this while doodling out the pars of some courses I know the par orders of, like Doral (544344453545343444), Augusta (454343454443545344), Pebble Beach (454435344443454435), St. Andrews (444454434434454444). Interestingly, Augusta isn't actually symmetrical but each nine at Augusta is symmetrical. If we require a nine to be symmetrical and to have the standard 2/5/2 breakdown, then it gets a little complicated. There are really only five slots to worry about, but you actually can't put anything but a par-4 in the 5th hole, because if you did you'd have an odd number of that kind of hole. By similar logic, you can't have a symmetric 3/3/3 nine. So in fact there are just twelve standard symmetric nines, and only nine of them are superstandard symmetric:

534444435
543444345
544343445
354444453
453444354
454343454 - Augusta front 9
345444543
435444534
445343544
344545443
434545434
443545344 - Augusta back 9

Anyway, that's my little fiddling around with golf course combinatorics.