# Baseball Puzzler: For extra credit

I thought the recent baseball puzzler was pretty good. I was puzzled for a while, but then I thought "hey, if the first half batting avg was .25 and the second half was .375, about the only thing you can say is that the entire season’s batting average was something in-between .25 and .375. " Without knowing how many at bats in each half, you can’t say more than that. So my mind finally un-puzzled, I was able to figure it out.

In Ray and Tom’s example, the number of at bats was in the hundreds. And some of the segment at-bats were half-at-bats; i.e. not integers. But an integer number of at bats would be necessary for actual baseball players.

Ok, try this for extra credit: What is the minimum number of at-bats consistent with the puzzler? Include the restriction that the number of at-bats in each segment by both players are all integers.

Remember, both players had the same total at bats for the seaon.

40 and 20 would result in a tie.

I figured at 40 too. Good on you!

Interesting a tie is possible. I didn’t look for ties. Just curious, how would 20 hit tie break-down, each hitter, in each segment?

.250 x 20 = 5
.375 x 40 = 15
total = 20

.300 x 40 = 12
.400 x 20 = 8
total = 20

The 40 and 20 were for the half seasons, 60 for the whole season. For a win and to hit those exact batting averages, the numbers would have to be 80 and 20.

Take a look at my numbers @keith. Do you concur?

I think a “win” can be completed with integer numbers only, and with just 40 total seasonal at bats. As follows:

For the player who bats 250 in the first segment, and 375 in the second
0.25 * 8 =2 hits in first segment with 8 at bats
0.375* 32 = 12 hits in second segment in 32 at bats
SeasonalTotal: 14 hits in 40 at bats

For the player who bats 300 in the first segment, and 400 in the second
0.300 * 30 = 9 hits in first segment with 30 at bats
0.400 * 10 = 4 hits in second segment with 10 at bats
Seasonal Total: 13 hits in 40 at bats

It appears to work, I was looking at 40 being the minimum for a batting average increment of .025 but it looks like 32 will do.