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And now for something a little different…

(This is adapted from a Car Talk puzzler.)

Bob and Sally are ball players. They both show up for spring training (they play in the Bluehorse pro league) and Bob, as ever, is boastful.

“Hey Sally,” he says, “I hit better than you before the All Star Break.”

“Yeah, I was injured in April,” says Sally. “You didn’t have to hit that well to beat me.”

“Well, the point is, I had a higher average than you did before the All Star Break. And then after the All Star Break, you went on a tear, you were hitting a lot more, getting on base a lot more. And yet I still had a higher average than you, since the All Star Break. Even though, you know, I was tired from being on the All Star Team.”

“Yeah,” says Sally, “I remember you playing a couple innings. I started at second base.”

“Well, the point is, I had a better average before the break, and a better average after the break. So I’m the better hitter.”

“Not so fast, Big Bob. I had the better average overall.”

“That can’t be,” says Bob. “I had the better average both halves of the season. And by the way, we had the exact same number of bats over the course of the season. How could your overall average possibly be higher than mine, if I beat you both halves, and we had the same number of times at the plate?”

“Nonetheless it’s true,” says Sally. And she shows him the stat book.

“How the heck can that be? Oh,” he says, and while Sally goes out and hits fungos, Bob’s gloomily checking the numbers for the fourteenth time.

So could this actually happen? Is it possible for one player to have a higher batting average than another both halves of the season, and yet the other player actually has a higher batting average for the season?