(Analysis by Nick Wu)

The $N$ hay bales define $N-1$ intervals that Bessie can be inside. Let's consider answering for a given interval, whether Bessie can escape if she starts inside that interval.

We can, in fact, ask a more general question - if Bessie is trapped between haybale $i$ and haybale $j$, can she escape? Clearly, if Bessie can break through haybale $i$, it is to her advantage to do so immediately - she then gains more distance and can possibly break through haybale $j$. However, if Bessie can break through neither haybale $i$ nor haybale $j$, then she is trapped.

We can then simulate this process as follows. Start by having Bessie be trapped between haybale $i$ and haybale $i+1$. While she still has a haybale to her left and a haybale to her right, see if she can break either one. Keep on breaking haybales until either she doesn't have one to her left or one to her right, or she can't break through either one. If she can't break through the haybales to her left and to her right, then take the distance of the original interval and add to that a running total. Repeat this simulation for all adjacent pairs of haybales.

Here is my Java code simulating this process.

import java.io.*;
import java.util.*;
public class trappedB {
public static void main(String[] args) throws IOException {
PrintWriter pw = new PrintWriter(new BufferedWriter(new FileWriter("trapped.out")));
Haybale[] bales = new Haybale[n];
for(int i = 0; i < n; i++) {
int size = Integer.parseInt(st.nextToken());
int position = Integer.parseInt(st.nextToken());
bales[i] = new Haybale(size, position);
}
// sort haybales by location
Arrays.sort(bales);
int ans = 0;
for(int i = 0; i < n-1; i++) {
int areaOfInterval = bales[i+1].position - bales[i].position;
int leftmostBale = i;
int rightmostBale = i+1;
// while Bessie could still be trapped
while(leftmostBale >= 0 && rightmostBale <= n-1) {
boolean broke = false;
int currDist = bales[rightmostBale].position - bales[leftmostBale].position;
if(currDist > bales[leftmostBale].size) {
leftmostBale--;
broke = true;
}
if(currDist > bales[rightmostBale].size) {
rightmostBale++;
broke = true;
}
// Bessie couldn't break through either the left or the right bale, so stop
if(!broke) {
break;
}
}
// Bessie couldn't break out
if(leftmostBale >= 0 && rightmostBale <= n-1) {
ans += areaOfInterval;
}
}
pw.println(ans);
pw.close();
}

static class Haybale implements Comparable<Haybale> {
public int position, size;
public Haybale(int sizeIn, int positionIn) {
size = sizeIn;
position = positionIn;
}
public int compareTo(Haybale h) {
// this will sort haybales from left to right
return position - h.position;
}
}

}