Problem:
Score: 300 points

Problem Statement

We have a 3 \times 3 grid. A number c_{i, j} is written in the square (i, j), where (i, j) denotes the square at the i-th row from the top and the j-th column from the left.

According to Takahashi, there are six integers a_1, a_2, a_3, b_1, b_2, b_3 whose values are fixed, and the number written in the square (i, j) is equal to a_i + b_j.

Determine if he is correct.

Constraints

c_{i, j} \ (1 \leq i \leq 3, 1 \leq j \leq 3) is an integer between 0 and 100 (inclusive).

Input

Input is given from Standard Input in the following format:

c_{1,1} c_{1,2} c_{1,3}
c_{2,1} c_{2,2} c_{2,3}
c_{3,1} c_{3,2} c_{3,3}

Output

If Takahashi's statement is correct, print Yes; otherwise, print No.

Sample Input 1

1 0 1
2 1 2
1 0 1

Sample Output 1

Yes

Takahashi is correct, since there are possible sets of integers such as: a_1=0,a_2=1,a_3=0,b_1=1,b_2=0,b_3=1.

Sample Input 2

2 2 2
2 1 2
2 2 2

Sample Output 2

No

Takahashi is incorrect in this case.

Sample Input 3

0 8 8
0 8 8
0 8 8

Sample Output 3

Yes

Sample Input 4

1 8 6
2 9 7
0 7 7

Sample Output 4

No

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

import java.util.HashMap;
import java.util.Map;
import java.util.Scanner;

public class Main {

	private static final String YES = "Yes";

	private static final String NO = "No";

	public static void main(String[] args) {

		int[][] arr = scan();
		System.out.println(process(arr) ? YES : NO);
	}

	private static int[][] scan() {

		Scanner sc = new Scanner(System.in);
		int[][] arr = new int[3][3];
		for (int i = 0; i < 3; i++) {
			for (int j = 0; j < 3; j++) {
				arr[i][j] = sc.nextInt();
			}
		}
		sc.close();
		return arr;
	}

	private static boolean process(int[][] arr) {

		Map<String, Integer> map = new HashMap<>();

		return invoke(arr, map, 0);
	}

	private static boolean invoke(int[][] arr, Map<String, Integer> map, int j) {

		while (j < 3) {

			String aKey = "a" + j;
			int a = arr[0][j];
			if (map.get(aKey) == null) {
				map.put(aKey, a);
			} else {
				a = map.get(aKey);

				if (a < 0) {
					return false;
				}
			}

			int b;
			String bKey;
			for (int i = 0; i < arr.length; i++) {
				bKey = "b" + i;
				b = arr[i][j] - a;

				if (map.get(bKey) == null) {
					map.put(bKey, b);
				} else if (b != map.get(bKey)) {
					if (i > 0) {
						map.put(aKey, map.get(aKey) - 1);
						return invoke(arr, map, j);
					}
					map.put(bKey, b);
				} else {
					// keyに紐づくvalueを使用する
					b = map.get(bKey);
				}

				if (b < 0 || arr[i][j] != (a + b)) {
					map.put(aKey, map.get(aKey) - 1);
					return invoke(arr, map, j);
				}
			}
			return invoke(arr, map, j + 1);
		}

		return true;
	}
}