How to Count

The other day I saw a math question disguised as a baseball trivia question. Here it is:

How many states don’t have major league baseball teams?

Let’s see: there’s Alaska, Arkansas,… Sure, it might be hard to list them all, but why am I calling this a math question?

Well, it doesn’t ask us to list all the states without baseball teams, it asks us to count them. Of course you can count things directly, by listing every one, but that’s not always as easy as it might seem. Maybe you can list all 50 states off the top of your head, and keep track as you go along of which ones don’t have teams, but I’m pretty sure I’ll overlook a few states. (I thought I could organize the states alphabetically, but I ended up giving up once I thought I got through the A’s, and I forgot Alabama!)

So how do you count things indirectly, without listing them? For a start, let’s reframe the question:

How many states DO have major league baseball teams?

If we can answer one question, we can answer the other: if, say, 20 states out of 50 have teams, then 30 don’t. But doesn’t the question feel a little easier when you ask it this second way? Pause here with me for just a moment: why is that?

One reason is that relatively few states have teams, and the ones that do are likely to be the better known ones, so if you were going to try to count by listing, listing the states that have teams is probably easier than listing the ones that don’t. But the real reason the alternate formulation helps is that you don’t have to count by listing — at least not by listing states. You could count by listing teams.

The Red Sox play in Massachusetts — that’s one state. The Yankees play in New York — that’s a second. The Giants play in California — a third. The A’s play in California too, but we already counted that. And so on.

We can make this process a little more organized if we use the structure of the baseball leagues. There are 30 major league baseball teams and they are currently divided evenly into two leagues: 15 in the American League, 15 in the National. Each league has 3 divisions — East, Central, and West — and each division has 5 teams. In other words: 30 teams broken up into 6 divisions of 5.

Doesn’t it feel a lot easier to go through 6 divisions of 5 than to go through 50 states? Let’s do it. I write this off the top of my head, in real time:

AL East: Boston Red Sox (MA, 1), New York Yankees (NY, 2), Baltimore Orioles (MD, 3), Toronto Blue Jays (Canada, not a state), Tampa Bay Rays (FL,4)

AL Central: Kansas City Royals (MO, 5), Detroit Tigers (MI, 6), Cleveland Indians (OH, 7), Minnesota Twins (MN, 8), Chicago White Sox (IL, 9)

AL West: Oakland A’s (CA, 10), Houston Astros (TX, 11), Texas Rangers (TX, repeat state), California Angels (CA, repeat state), Seattle Mariners (WA, 12)

NL East: Washington Nationals (DC, not a state), New York Mets (NY, repeat state), Philadelphia Phillies (PA, 13), Miami Marlins (FL, repeat state), Atlanta Braves (GA, 14)

NL Central: St. Louis Cardinals (MO, repeat state), Pittsburgh Pirates (PA, repeat state), Milwaukee Brewers (WI, 15), Cincinnati Reds (OH, repeat state), Chicago Cubs (IL, repeat state)

NL West: Arizona Diamondbacks (AZ, 16), Colorado Rockies (CO, 17), San Diego Padres (CA, repeat state), Los Angeles Dodgers (CA, repeat state), San Francisco Giants (CA, repeat state)

And there you have it: 17 distinct states with teams, so 33 states without. And while this problem isn’t winning anybody the Fields Medal, it does illustrate two very important principles of counting, and math in general:

1. Find and use correspondences. When we asked which states have teams, we set up an implicit correspondence between states and teams. A way to make that correspondence more explicit is to reframe the question yet again, this time in terms of team-state pairs:

How many pairs (S, T) are there, where S is a state, T is a team that plays in that state, and no state is repeated more than once?

This might sound needlessly complicated, but math people actually like to talk this way! (Remember the definition of relations and functions the first time you saw it? Your eyes probably glazed over; mine sure did.) We use this language because it brings to the surface the duality inherent in the set-up: states and teams are paired. When you have pairs, you get to choose how to enumerate them: over the first entry, or over the second. And in this case, the second is the way to go, because…

2. More structure is better. The set of states seems sort of amorphous. You can try to break it up into regions (New England, Mid-Atlantic, Midwest,…), but it’s not totally clear how to do it. Whereas the set of baseball teams has a very clear structure: six by five. I lied in one place when I told you I was listing baseball teams in real time. When I got to the NL Central, I put down three of the five teams, and then spaced on what the other two were. But I knew there had to be five, and I knew about where they should be geographically. I remembered the other two within a minute.

Counting has a rich and noble history. Also a fancier name: combinatorics. And while the subject, perhaps like much of math, might seem like a bag of tricks when you first encounter it, it has some clear guiding principles. Look for structures, and try to transform your problem so you can make use of those structures. These principles are at work all over, so keep an eye out for them!

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