Problem:
Score: 300 points

Problem Statement

In the Ancient Kingdom of Snuke, there was a pyramid to strengthen the authority of Takahashi, the president of AtCoder Inc.

The pyramid had center coordinates (C_X, C_Y) and height H. The altitude of coordinates (X, Y) is max(H - |X - C_X| - |Y - C_Y|, 0).

Aoki, an explorer, conducted a survey to identify the center coordinates and height of this pyramid. As a result, he obtained the following information:

C_X, C_Y was integers between 0 and 100 (inclusive), and H was an integer not less than 1.

Additionally, he obtained N pieces of information. The i-th of them is: "the altitude of point (x_i, y_i) is h_i."

This was enough to identify the center coordinates and the height of the pyramid. Find these values with the clues above.

Constraints

N is an integer between 1 and 100 (inclusive).

x_i and y_i are integers between 0 and 100 (inclusive).

h_i is an integer between 0 and 10^9 (inclusive).

The N coordinates (x_1, y_1), (x_2, y_2), (x_3, y_3), ..., (x_N, y_N) are all different.

The center coordinates and the height of the pyramid can be uniquely identified.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1 h_1
x_2 y_2 h_2
x_3 y_3 h_3
:
x_N y_N h_N

Output

Print values C_X, C_Y and H representing the center coordinates and the height of the pyramid in one line, with spaces in between.

Sample Input 1

4
2 3 5
2 1 5
1 2 5
3 2 5

Sample Output 1

2 2 6

In this case, the center coordinates and the height can be identified as (2, 2) and 6.

Sample Input 2

2
0 0 100
1 1 98

Sample Output 2

0 0 100

In this case, the center coordinates and the height can be identified as (0, 0) and 100.

Note that C_X and C_Y are known to be integers between 0 and 100.

Sample Input 3

3
99 1 191
100 1 192
99 0 192

Sample Output 3

100 0 193

In this case, the center coordinates and the height can be identified as (100, 0) and 193.

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.ArrayList;
import java.util.HashMap;
import java.util.HashSet;
import java.util.List;
import java.util.Map;
import java.util.Scanner;
import java.util.Set;

public class Main {
	public static void main(String[] args) {
		Scanner scan = new Scanner(System.in);
		int N = scan.nextInt();
		int[]x = new int[N];
		int[]y = new int[N];
		int[]h = new int[N];
		for(int i = 0; i < N; i++) {
			x[i] = scan.nextInt();
			y[i] = scan.nextInt();
			h[i] = scan.nextInt();
		}
		scan.close();

		ArrayList<Map<Integer, ArrayList<int[]>>> mlist = new ArrayList<Map<Integer, ArrayList<int[]>>>();
		for(int i = 0; i < N; i++) {
			Map<Integer, ArrayList<int[]>> map = new HashMap<Integer, ArrayList<int[]>>();
			for(int cx = 0; cx <= 100; cx++) {
				for(int cy = 0; cy <= 100; cy++) {
					int H = h[i] + Math.abs(x[i] - cx) + Math.abs(y[i]- cy);
					if(H == 0) continue;
					if(map.containsKey(H)) {
						ArrayList<int[]> list = map.get(H);
						int[]t = {cx, cy};
						list.add(t);
						map.put(H, list);
					}else {
						int[]t = {cx, cy};
						ArrayList<int[]> list = new ArrayList<int[]>();
						list.add(t);
						map.put(H, list);
					}
				}
			}
			mlist.add(map);
		}
		Set<Integer> set = new HashSet<Integer>();
		for(int k : mlist.get(0).keySet()){
			boolean flag = true;
			for(int i = 1; i < N; i++) {
				Map<Integer, ArrayList<int[]>> map = mlist.get(i);
				if(!map.containsKey(k)) {
					flag = false;
					break;
				}
			}
			if(flag) {
				if(k >= 1 && k <= 100) {
					set.add(k);
				}
			}
		}
		int cnt = 0;
		int a[] = new int[3];
		List<int[]> list = new ArrayList<int[]>();
		for(int i : set) {
			Map<Integer, ArrayList<int[]>> map = mlist.get(0);
			for(int j = 0; j < map.get(i).size(); j++) {
				int[]p = map.get(i).get(j);
				int cx = p[0];
				int cy = p[1];
				//System.out.println(cx + " " + cy);
				boolean flag1 = true;
				for(int k = 0; k < N; k++) {
					int t = Math.max(0, i - Math.abs(x[k] - cx) - Math.abs(y[k] - cy));
					if(t != h[k]) {
						flag1 = false;
						break;
					}
				}
				if(flag1) {
					boolean flag0 = true;
					for(int k = 1; k < N; k++) {
						Map<Integer, ArrayList<int[]>> mapK = mlist.get(k);
						List<int[]> listK = mapK.get(i);
						boolean flag = false;
						for(int l = 0; l < listK.size(); l++){
							int[]pK = listK.get(l);
							if(cx == pK[0] || cy == pK[1]) {
								break;
							}
						}
						if(flag) {
							flag0 = false;
							break;
						}
					}
					if(flag0) {
						System.out.println(cx + " " + cy + " " + i);
						System.exit(0);
					}
				}
			}
		}
	}
}