Mathematical Baseball

This week I have been noticing that of all the team sports, especially those that are broadcast extensively, baseball is the most mathematical. It is a discrete game; that is, the game proceeds by innings, which are composed of outs, which are composed of trips to the plate, which are composed of discrete pitches. It is almost as though a pitch was an "atom" of baseball. Baseball games are composed into box scores, which total the number of items that occurred in the game by categories. For example, you will see the number of runs by each team, the number of hits by each player, the number of walks allowed by each pitcher, and so forth, and from these statistics can be developed, such as standings and batting averages. One can even simulate baseball based on what has happened in previous games.

One of my main interests is what can happen in baseball? What are the logical extremes? For example:

1. What is the fewest number of pitches in a game?

This one is a little tricky. Think about it a while and then come back to this blog.

Did you say 54 pitches? Each player bangs into an out on the very first pitch, 3 pitches a half inning, and so 3 x 2 x 9 = 54 pitches in a game? Almost, but not quite. If the home team is ahead, the bottom of the 9th will not be played, and that makes it 51 pitches. But that's not correct, either. If it were 51 pitches, every one would be an out, and the score would be tied, 0-0. One more pitch is needed somewhere to score a run for the home team, probably a solo home run. So the answer is 52.

2. What is the most number of pitches in a game?

The answer is infinity. This gets into a concept I have invented in games called entropy. In games, some moves are reversible; you can go back to the same situation again. Others are irreversible. In chess, for example, moving a knight to an open square is reversible; you can move him back again. But capturing a piece is irreversible, as the piece can never return, and moving a pawn is irreversible, as pawns can't move backward. And so it is with baseball. An out is irreversible; one can't undo the out and go back from 1 out to none, for instance. A hit, however, is reversible, since a team and hit and hit and hit over again and the situation will still remain the same, especially if the hits are home runs. Conceivably the team could score run after run after run as the New York Yankees did yesterday (against Tampa Bay on 2005 June 21) without ever getting the third out. So the game goes infinite. (Well actually, the Yankees stopped after 13 runs) There are four ways a game can go infinite:

a. Run infinite. A team scores run after run after run in an inning without the third out ever coming.

b. Inning infinite. The game goes into extra innings, one after another, for an infinite amount of time because the two teams score the same number of runs in each inning.

c. Foul ball infinite. A foul ball after two strikes (unless it is a tip or is bunted) does not count anything in baseball - it's reversible. So have a batter hit foul after foul after foul without ever causing an out or hitting and getting safe on first. There is even a book written about this idea, "The Boy who Batted 1.000".

d. Pickoff infinite. The pitcher over and over again attempts to pick off a runner with a throw to that base, and with the runner making it back on time each time. There may be a rule against this one, something about delay of game.

If one assumes that none of these infinites can occur, then a baseball game must end. However, the maximum number of pitches depends on the limits you place on innings, batter-friendly events, foul balls, and pickoff attempts.

3. Suppose a pitcher gets 7 strikeouts in a single inning. Show that a run must have scored.

This one is strange. How can a pitcher get more than 3 strikeouts? 3 outs ends the inning. The answer is that if the catcher can't catch the ball, and there are two outs or first base is open, then the batter who is struck out can run for first base, and if he beats the ball, he is safe at first. The pitcher still gets a strikeout. So 7 strikeouts means 7 people to the plate, and one can fit only 3 of these in outs and 3 on the bases, so at least one run must have scored.

4. How many hits must a team get before a team is certain of scoring a run?

You think it may be something like 10? 15? 22? Try 55. That's right. A team must get 55 or more hits in a game before it becomes certain that a run must have occurred. A team can get 54 hits, and furthermore, 27 of these can be triples, without scoring a single run. Here's how:

A. Hits triple and is out trying to score.
B. Hits triple and is out trying to score.
C. Hits triple.
D. Hits single; 3rd base coach holds runner at 3rd.
E. Hits single; 3rd base coach holds runner at 3rd.

Here is where the problem occurs. If the next runner is out, that ends the inning with only 5 hits. If the next runner hits, with the bases loaded, a run must score. It seems like we need a hit and an out simultaneously and how do we get that? Here's how:

F. Hits ball, batted ball hits a runner. F is credited with a single. The runner is automatically out.

And there it is, 6 hits and no runs. Repeat that over 9 innings and you get 54 hits, with 27 triples, and no runs. We can fire this third base coach.

5. Can a pitcher lose a no-hitter?

Yes he can. A pitcher must win a shutout, because you need runs to win a game. A pitcher must win a perfect game, for you must get on base before you can score. But a pitcher can very well lose a no-hitter. Hits are not necessary for a win. All you need is 3 errors, as happened on 1990 July 1 with a game between the New York Yankees and the Chicago White Sox. The Yankees had 4 hits but 0 runs, and the White Sox had 0 hits but 4 runs. The no-hit White Sox won the game by a substantial margin.