I think this is all you need to know to solve: bases are numbered 0,1,2,3; batter starts at 0 (home plate) and becomes runner traveling in order if result is a single; returning to 0 after 3 scores a run and runner is removed; home run scores batter and all runners and all are removed; team gets 3 outs per inning; lineup is 1-9 batting in order and returns to 1 after 9
I agree. For the sake of the extra credit, my hangup is whether the batting order continues where it left off with each inning or if it resets to the first batter. I'm pretty sure it continues, but confirmation would be nice.
When I first saw the "Making the Rounds," I misread it and thought it was about iteratively taking means-of-means-of-means, etc. So for instance, you could find an iterated average of the set {1,2,3,4,5} in many ways, including (((1+2+3)/3+4)/2+5)/2.
In that case, the problem turns out to be pretty easy. You can maximize the average of {1...n} by taking ((...(1+2)/2+3)/2+...+n)/2, since that gives maximum weight to the biggest terms. Some quick analytical arguments (to find the indefinite sum of k x^(k-1)) or arguments by induction show that this expression equals (n–1)/2 + 2^(1–n) for all positive integers n, which exceeds the arithmetic mean of (n+1)/2 by (n–3)/2 + 2^(1–n).
For n = 12, that gives an average of 11 + 1/2048 = 11.00048828125, exceeding the arithmetic mean of 6.5 by 4.50048828125.
Here's an abridged version that's relevant for this puzzle:
A team has nine batters. First comes batter 1, then 2, then 3, and so on, through batter 9, after which the cycle repeats with batter 1, and so on. There are three bases: first, second, and third. Upon hitting a single, the batter goes to first base. When the next batter hits a single, that first batter advances to second base. And then third. After third base, the batter runs to "home plate" and scores a run for the team at that point.
If a batter records an out, they do not get on base, and the next batter is up. In real baseball, there are ways for runners (batters who are on base) to advance to the next base when there's an out, but I have intentionally excluded this from the puzzle. In each inning, once a team collectively earns three outs, the inning is over. When they resume batting in the next inning, the bases are once again initially empty, and the next batter in the order is up.
When a batter hits a home run, that batter and all the runners currently on base score a run -- that is, they all cross home plate, and the bases are once again empty for the next batter.
I won! Hooray!
A pointer to the minimum set of rules of baseball needed to solve this problem would be nice.
I think this is all you need to know to solve: bases are numbered 0,1,2,3; batter starts at 0 (home plate) and becomes runner traveling in order if result is a single; returning to 0 after 3 scores a run and runner is removed; home run scores batter and all runners and all are removed; team gets 3 outs per inning; lineup is 1-9 batting in order and returns to 1 after 9
I agree. For the sake of the extra credit, my hangup is whether the batting order continues where it left off with each inning or if it resets to the first batter. I'm pretty sure it continues, but confirmation would be nice.
That's correct, the batting order continues. (It does not restart at 1 after each inning.)
When I first saw the "Making the Rounds," I misread it and thought it was about iteratively taking means-of-means-of-means, etc. So for instance, you could find an iterated average of the set {1,2,3,4,5} in many ways, including (((1+2+3)/3+4)/2+5)/2.
In that case, the problem turns out to be pretty easy. You can maximize the average of {1...n} by taking ((...(1+2)/2+3)/2+...+n)/2, since that gives maximum weight to the biggest terms. Some quick analytical arguments (to find the indefinite sum of k x^(k-1)) or arguments by induction show that this expression equals (n–1)/2 + 2^(1–n) for all positive integers n, which exceeds the arithmetic mean of (n+1)/2 by (n–3)/2 + 2^(1–n).
For n = 12, that gives an average of 11 + 1/2048 = 11.00048828125, exceeding the arithmetic mean of 6.5 by 4.50048828125.
Here's an abridged version that's relevant for this puzzle:
A team has nine batters. First comes batter 1, then 2, then 3, and so on, through batter 9, after which the cycle repeats with batter 1, and so on. There are three bases: first, second, and third. Upon hitting a single, the batter goes to first base. When the next batter hits a single, that first batter advances to second base. And then third. After third base, the batter runs to "home plate" and scores a run for the team at that point.
If a batter records an out, they do not get on base, and the next batter is up. In real baseball, there are ways for runners (batters who are on base) to advance to the next base when there's an out, but I have intentionally excluded this from the puzzle. In each inning, once a team collectively earns three outs, the inning is over. When they resume batting in the next inning, the bases are once again initially empty, and the next batter in the order is up.
When a batter hits a home run, that batter and all the runners currently on base score a run -- that is, they all cross home plate, and the bases are once again empty for the next batter.