The Logic of Baseball

Introducing a new kind of puzzle

Welcome to the many new subscribers who joined after the last newsletter went semi-viral. I tend to write about something different every issue, so if this one isn’t your cup of tea, stick around to see what other interesting corners of culture I find to share with you. And be sure to check out the companion video series on YouTube. But now, let’s play ball!

Apparently, it’s baseball season. I say “apparently” because I remember there’s a book about baseball called The Boys of Summer, and summer starts soon, but I just looked up when baseball season starts and it turns out we’re already about three months into it and I had no idea.

All of this is to say that I’m not really much of a sports guy. I enjoy individual games of most sports, but I don’t follow teams or players or know things like when the seasons begin.

But I do like a good puzzle. And my father-in-law Jerry is one of the most brilliant puzzlers I’ve ever met, the kind of guy who excels at anything requiring math or logic. Extremely long-time readers might remember way back when the popular word game SpellTower came out, and the highest score on the leaderboard was 167,275 but Jerry came along and shattered that score with more than a million points.

Jerry’s friend Jim is similarly puzzle-minded. And together they have come up with a new kind of logic puzzle about baseball. It’s perfect for someone like me who knows the basic rules of baseball (there are 9 players on a team, 9 innings per game, 3 strikes per out, etc) and likes logic puzzles, and doesn’t require knowing anything beyond that.

The concept: Given a specific snapshot of a baseball game, can you use logic to answer a question about the game?

Consider this scenario:

Casey, batting third for the Mudville Slugs, came to the plate in the ninth inning for his fourth at-bat. The bases were loaded. There were two outs, and his team was two runs behind. He struck out, ending the game. What was the final score?

Do you think there’s enough information there to answer the question? At first, I was baffled about how to even approach this. But when I heard their explanation, it all clicked. Let’s walk it through together.

We know that everyone who appears at the plate either gets out, scores a run, or is left on base at the end of the inning. So you can write this as a simple formula:

Plate Appearances = Outs + Runs + Left on base

The scenario told us that Casey was third in the batting order and this was his fourth time at bat.

For Casey to be at bat a fourth time, they must have gone through the full batting order three times. With nine players on a team, three full rotations means 27 appearances at the plate. And since Casey is third in the batting order, that adds three plate appearances from the fourth rotation for a total of 30 when the game ended.

Now that we know the total number of Plate Appearances, we can update the formula:

30 = Outs + Runs + Left on base

We also know from the scenario that this is the end of the ninth inning. With three outs per inning, that’s 27 outs. So we can update it again:

30 = 27 + Runs + Left on base

The scenario says the bases are loaded. When Casey strikes out and ends the game, that leaves three players left on base. So now we can fill in that number, too:

30 = 27 + Runs + 3

Now, it’s possible there were players left on base in other innings, but not in this case since the combination of Outs and Left On Base is already equal to the Plate Appearances. And the only way this works mathematically is if Runs = 0, so now we know Mudville scored no runs. And since “his team was two runs behind,” that makes the final score 2 - 0!

Okay, I get it. That’s pretty clever!

The idea for this type of puzzle began with Jerry reading box scores in the newspaper. That’s a condensed way of showing the entire outcome of a baseball game in a neat little grid. And Jerry became fascinated with what he could figure out about a game just by looking at the box scores in the newspapers.

He shared his fascination with Jim, and together they have written an entire book of baseball logic puzzles based on what you can figure out from box scores.

It’s written as a dialogue between two fictional characters who happen to be named Jim and Jerry. It’s extremely readable and entertaining. Here is an excerpt that introduces the idea of box scores:

The Game Begins

Jerry: Jim, we’ve been great friends for more than fifty years, and we know each other very well, but I have a hobby that I don’t think I’ve ever told you about. Every day in the baseball season, I like to study the box scores of the games. I like to see how my favorite teams and players are doing, but that’s not what my hobby is about. Instead, I study the box scores to look for the hidden information they contain.

Jim: Wow. Fifty years.

What do you mean, “hidden information?”

Jerry: Well, the box score only presents aggregate information about the game. For example, it tells us for each player how many at-bats, hits, runs, and runs batted in he got in the game. But it doesn’t say in which at bat or which inning he got his hits and scored his runs. The box score might tell us that the visiting team rallied to tie the game in the top of the ninth, but it doesn’t tell us who scored in the ninth and who drove in the tying run. To be precise, it doesn’t tell us explicitly. But often we can deduce this more granular information. Often we can recover some of the specific events in the game that were used to create the box score.

Jim: Whoa! That sounds really interesting! I never thought of that before.

But ... what can you deduce? Can you give me an example?

Jerry: Consider this box score, which I made up:

[note: annotations in red added by David]

Jerry: As you can see, the Cats scored runs in the first and eighth innings. The box score lists the batters according to the order in which they hit. We can see that Castro and Dominguez each scored a run, and Dominguez and Engel each batted in a run. But suppose we want to know who scored the winning run in the eighth, and who drove it in. See if you can figure that out.

Jim: Ooh! Can I tell who is up in the eighth? Umm... I don’t think so. Total at-bats is 36... but that doesn’t count walks.

Hey, I’m getting nowhere.

Jerry: Look at the first inning instead.

Jim: Oh. Well, either Castro or Dominguez scores. Couldn’t it be either one? I mean, let’s see, first Abe is up. I guess he gets out. And then so does Brown. And then Castro could get on. And Dominguez could drive him in. So Castro could score in the first.

And could Dominguez score instead? Well sure, because Abe and Brown are out, then Castro... can’t get out, that would end the inning, so... but if he gets on…

Oh I see! Dominguez can’t score in the first, because Abe, Brown, and Castro would all have to get out before Dominguez can score and that would end the inning!

So it must be Dominguez who scores in the eighth!

Jerry: And who drove him in?

Jim: Well, Engel of course. He’s the only one who could.

Jerry: Couldn’t Dominguez drive himself in?

Jim: Oh. Well. I guess he could. He could hit a home run.

Why are you looking at me like that?

Oh. No, he didn’t hit a homer. If he did, it would say so at the bottom of the box score! So Engel drove him in. And Dominguez drove in Castro in the first.

Wow. That was tricky!

Jerry: Well, sort of. But all you really needed was a basic understanding of the game.

Baseball has lots of rules. Most rules are so well-understood that you don’t notice that you’re using them. For example, in figuring out the puzzle, you understood that if Castro gets on base with 2 outs in the first inning, then Dominguez can't score. To score, without Castro getting out or scoring first, Dominguez would have to pass Castro on the basepaths. He would be called out for that.

Jim: Ah! Right. I’m sure I was thinking of that.


Jim: But Jerry, what’s the point? You can get the complete play-by-play summary of any game on!

Jerry: But that misses the fun! The fun is in figuring out a puzzle!

So great! If you like puzzles, logic, and/or baseball, you should check out their book. There are puzzles based on imagined games, real life historic games, little league scenarios, and so on. It’s called The Baseball Mysteries: Challenging Puzzles for Logical Detectives and it’s available in paperback or kindle versions on Amazon.

And that’s it for another newsletter. Thanks as always for reading, and a special thank you to those of you just joining us! Stick around for more!


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