(Analysis by Nick Wu)

In this problem, Bessie and Elsie each have $N$ unique integers ranging from 1 to $2N$. Over $N$ rounds, Bessie and Elsie will each select a different integer and the cow who selected the larger integer wins a point. Bessie knows what integers Elsie will select and wants to maximize her (Bessie's) score.

Because Bessie knows the exact order in which Elsie will select her integers, Bessie can determine her exact strategy beforehand, choosing to win points on certain rounds and not win points on other rounds.

Assume without loss of generality that Bessie is able to earn $K$ points using some strategy. We claim that there exists a strategy where Bessie wins points exactly when Elsie picks her $K$ smallest integers. The intuition for this is as follows - if there are two integers $x$ and $y$ where $x<y$, Elsie picks $x$ and Bessie loses, and Elsie picks $y$ and Bessie wins, then Bessie can swap the two moves - Bessie will win when Elsie picks $x$ and lose when she picks $y$.

Therefore, it remains to see how many of Elsie's smallest integers Bessie can beat.

Bessie can start by trying to beat Elsie's smallest integer. If none of Bessie's integers are larger than that one, Bessie can't win any points. Otherwise, Bessie needs to pick an integer to beat Elsie's smallest integer. Intuitively, the best choice is to pick the smallest integer that can beat it - the reason for this is that we want to save bigger integers for later on.

After beating Elsie's smallest integer, Bessie can then try to beat her second smallest one, and continue beating Elsie's integers in order until she can no longer beat one.

To efficiently find the smallest such integer that can beat a given one from Elsie, we can consider Bessie's integers in order. If one of her integers can't beat Elsie's smallest unbeaten integer, it will never be usable and we can ignore it. Otherwise, we use that integer to beat Elsie's integer.

Here is my code illustrating this:

import java.io.*;
import java.util.*;
public class highcard {
public static void main(String[] args) throws IOException {
PrintWriter pw = new PrintWriter(new BufferedWriter(new FileWriter("highcard.out")));
int n = Integer.parseInt(br.readLine());
boolean[] elsieOwns = new boolean[2*n+1];
for(int i = 0; i < n; i++) {
}
ArrayList<Integer> bessie = new ArrayList<Integer>();
ArrayList<Integer> elsie = new ArrayList<Integer>();
int points = 0;
// because we loop over the values in increasing order, the two lists will be in sorted order
for(int i = 1; i <= 2*n; i++) {
if(elsieOwns[i]) {
}
else {
}
}
int bessieIndex = 0;
int elsieIndex = 0;
while(bessieIndex < n && elsieIndex < n) {
if(bessie.get(bessieIndex) > elsie.get(elsieIndex)) {
points++;
bessieIndex++;
elsieIndex++;
}
else {
bessieIndex++;
}
}
pw.println(points);
pw.close();
}
}