从 NxN 棋盘上的起点打印所有可能的骑士之旅
给定一个N x N棋盘，骑士最初站在第X行和第Y列，任务是打印所有可能的路径，使得骑士必须恰好访问每个方格一次。
例子：
Input: N = 5, X = 1, Y = 1
Output:
1 6 15 10 21
14 9 20 5 16
19 2 7 22 11
8 13 24 17 4
25 18 3 12 23
1 6 11 18 21
12 17 20 5 10
7 2 15 22 19
16 13 24 9 4
25 8 3 14 23
… 302 more
Explanation: Initially, the knight is at (1, 2)th cell. According to the 1st path, the knight will visit the cells in the following order: (1, 1) -> (3, 2) -> (5, 3) -> (4, 5) -> (2, 4) … and so on.
Input: N = 3, X = 1, Y = 3
Output: -1
Explanation: There exist no valid sequence of path such that the knight visit each square exactly once.
方法：这个问题可以通过递归和回溯的帮助来解决，方法是一一生成所有可能的旅行并检查它是否满足给定的条件。 Knight's Tour Problem 中讨论了对类似方法的更彻底的解释。以下是要遵循的步骤：
- 创建一个递归函数来迭代骑士可以遵循的所有可能路径。
- 使用变量visited维护Knight 访问的方格数。
- 创建一个函数isSafe() ，该函数将正方形的坐标作为参数并返回该正方形是否对骑士的下一次移动有效。
- 遍历骑士所有可能的 8 个动作，并检查它们是否可以安全访问。如果是，则递归调用下一步，直到访问的方格数等于方格总数。
时间复杂度： O(8 N*N )
辅助空间：O(1)
下面是上述方法的实现：
C++
// C++ program of the above approach
#include
using namespace std;
// Stores the 8 possible combinations of
// moves that the knight can follow
int DirX[] = { 2, 1, -1, -2, -2, -1, 1, 2 };
int DirY[] = { 1, 2, 2, 1, -1, -2, -2, -1 };
// Function to find if (i, j) is a valid
// cell for the knight to move and it
// exists within the chessboard
bool isSafe(int i, int j, int n,
vector >& Board)
{
return (i >= 0 and j >= 0 and i < n and j < n
and Board[i][j] == 0);
}
// Stores whether there exist any valid path
bool isPossible = false;
// Recursive function to iterate through all
// the paths that the knight can follow
void knightTour(vector >& ChessBoard, int N,
int x, int y, int visited = 1)
{
// Mark the current square of the chessboard
ChessBoard[x][y] = visited;
// If the number of visited squares are equal
// to the total number of squares
if (visited == N * N) {
isPossible = true;
// Print the current state of ChessBoard
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
cout << ChessBoard[i][j] << " ";
}
cout << endl;
}
cout << endl;
// Backtrack to the previous move
ChessBoard[x][y] = 0;
return;
}
// Iterate through all the eight possible moves
// for a knight
for (int i = 0; i < 8; i++) {
// Stores the new position of the knight
// after a move
int newX = x + DirX[i];
int newY = y + DirY[i];
// If the new position is a valid position
// recursively call for the next move
if (isSafe(newX, newY, N, ChessBoard)
&& !ChessBoard[newX][newY]) {
knightTour(ChessBoard, N, newX, newY,
visited + 1);
}
}
// Backtrack to the previous move
ChessBoard[x][y] = 0;
}
// Driver Code
int main()
{
vector > ChessBoard(5, vector(5, 0));
int N = ChessBoard.size();
int X = 1;
int Y = 1;
knightTour(ChessBoard, N, X - 1, Y - 1);
// If no valid sequence of moves exist
if (!isPossible) {
cout << -1;
}
return 0;
} Java
// Java program of the above approach
class GFG {
// Stores the 8 possible combinations of
// moves that the knight can follow
static int[] DirX = { 2, 1, -1, -2, -2, -1, 1, 2 };
static int[] DirY = { 1, 2, 2, 1, -1, -2, -2, -1 };
// Function to find if (i, j) is a valid
// cell for the knight to move and it
// exists within the chessboard
static boolean isSafe(int i, int j, int n, int[][] Board) {
return (i >= 0 && j >= 0 && i < n && j < n && Board[i][j] == 0);
}
// Stores whether there exist any valid path
static boolean isPossible = false;
// Recursive function to iterate through all
// the paths that the knight can follow
static void knightTour(int[][] ChessBoard, int N, int x, int y, int visited)
{
// Mark the current square of the chessboard
ChessBoard[x][y] = visited;
// If the number of visited squares are equal
// to the total number of squares
if (visited == N * N) {
isPossible = true;
// Print the current state of ChessBoard
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
System.out.print(ChessBoard[i][j] + " ");
}
System.out.println();
}
System.out.println();
// Backtrack to the previous move
ChessBoard[x][y] = 0;
return;
}
// Iterate through all the eight possible moves
// for a knight
for (int i = 0; i < 8; i++) {
// Stores the new position of the knight
// after a move
int newX = x + DirX[i];
int newY = y + DirY[i];
// If the new position is a valid position
// recursively call for the next move
if (isSafe(newX, newY, N, ChessBoard)
&& ChessBoard[newX][newY] == 0) {
knightTour(ChessBoard, N, newX, newY,
visited + 1);
}
}
// Backtrack to the previous move
ChessBoard[x][y] = 0;
}
// Driver Code
public static void main(String args[]) {
int[][] ChessBoard = new int[5][5];
int N = ChessBoard.length;
int X = 1;
int Y = 1;
knightTour(ChessBoard, N, X - 1, Y - 1, 1);
// If no valid sequence of moves exist
if (isPossible == false) {
System.out.println(-1);
}
}
}
// This code is contributed by Saurabh JaiswalC#
// C# program of the above approach
using System;
class GFG {
// Stores the 8 possible combinations of
// moves that the knight can follow
static int[] DirX = { 2, 1, -1, -2, -2, -1, 1, 2 };
static int[] DirY = { 1, 2, 2, 1, -1, -2, -2, -1 };
// Function to find if (i, j) is a valid
// cell for the knight to move and it
// exists within the chessboard
static bool isSafe(int i, int j, int n, int[, ] Board)
{
return (i >= 0 && j >= 0 && i < n && j < n
&& Board[i, j] == 0);
}
// Stores whether there exist any valid path
static bool isPossible = false;
// Recursive function to iterate through all
// the paths that the knight can follow
static void knightTour(int[, ] ChessBoard, int N, int x,
int y, int visited = 1)
{
// Mark the current square of the chessboard
ChessBoard[x, y] = visited;
// If the number of visited squares are equal
// to the total number of squares
if (visited == N * N) {
isPossible = true;
// Print the current state of ChessBoard
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
Console.Write(ChessBoard[i, j] + " ");
}
Console.WriteLine();
}
Console.WriteLine();
// Backtrack to the previous move
ChessBoard[x, y] = 0;
return;
}
// Iterate through all the eight possible moves
// for a knight
for (int i = 0; i < 8; i++) {
// Stores the new position of the knight
// after a move
int newX = x + DirX[i];
int newY = y + DirY[i];
// If the new position is a valid position
// recursively call for the next move
if (isSafe(newX, newY, N, ChessBoard)
&& ChessBoard[newX, newY] == 0) {
knightTour(ChessBoard, N, newX, newY,
visited + 1);
}
}
// Backtrack to the previous move
ChessBoard[x, y] = 0;
}
// Driver Code
public static void Main()
{
int[, ] ChessBoard = new int[5, 5];
int N = ChessBoard.GetLength(0);
int X = 1;
int Y = 1;
knightTour(ChessBoard, N, X - 1, Y - 1);
// If no valid sequence of moves exist
if (isPossible == false) {
Console.WriteLine(-1);
}
}
}
// This code is contributed by ukasp.Javascript
输出
1 6 15 10 21
14 9 20 5 16
19 2 7 22 11
8 13 24 17 4
25 18 3 12 23
1 6 11 18 21
12 17 20 5 10
7 2 15 22 19
16 13 24 9 4
25 8 3 14 23
1 6 11 16 21
12 15 20 5 10
7 2 13 22 17
14 19 24 9 4
25 8 3 18 23
1 6 17 12 21
16 11 20 5 18
7 2 9 22 13
10 15 24 19 4
25 8 3 14 23
1 12 17 6 21
18 5 20 11 16
13 2 9 22 7
4 19 24 15 10
25 14 3 8 23
1 16 11 6 21
10 5 20 15 12
17 2 13 22 7
4 9 24 19 14
25 18 3 8 23
1 18 11 6 21
10 5 20 17 12
19 2 15 22 7
4 9 24 13 16
25 14 3 8 23
1 10 15 6 21
16 5 20 9 14
11 2 7 22 19
4 17 24 13 8
25 12 3 18 23
1 16 5 10 21
6 11 20 15 4
19 2 17 22 9
12 7 24 3 14
25 18 13 8 23
1 12 5 18 21
6 17 20 13 4
11 2 9 22 19
16 7 24 3 14
25 10 15 8 23
1 10 5 16 21
6 15 20 11 4
9 2 7 22 17
14 19 24 3 12
25 8 13 18 23
1 18 7 12 21
8 13 20 17 6
19 2 5 22 11
14 9 24 3 16
25 4 15 10 23
1 6 17 12 21
18 11 20 7 16
5 2 15 22 13
10 19 24 3 8
25 4 9 14 23
1 6 15 12 21
16 11 20 7 14
5 2 13 22 19
10 17 24 3 8
25 4 9 18 23
1 12 17 8 21
18 7 20 3 16
13 2 11 22 9
6 19 24 15 4
25 14 5 10 23
1 16 13 8 21
12 7 20 3 14
17 2 15 22 9
6 11 24 19 4
25 18 5 10 23
1 18 13 8 21
12 7 20 3 14
19 2 17 22 9
6 11 24 15 4
25 16 5 10 23
1 10 15 8 21
16 7 20 3 14
11 2 9 22 19
6 17 24 13 4
25 12 5 18 23
1 4 15 10 21
14 9 20 3 16
19 2 5 22 11
8 13 24 17 6
25 18 7 12 23
1 4 9 18 21
10 17 20 3 8
5 2 13 22 19
16 11 24 7 14
25 6 15 12 23
1 4 9 16 21
10 15 20 3 8
5 2 11 22 17
14 19 24 7 12
25 6 13 18 23
1 4 17 12 21
16 11 20 3 18
5 2 7 22 13
10 15 24 19 8
25 6 9 14 23
1 16 3 10 21
6 11 20 15 4
17 2 5 22 9
12 7 24 19 14
25 18 13 8 23
1 18 3 12 21
8 13 20 17 4
19 2 7 22 11
14 9 24 5 16
25 6 15 10 23
1 8 3 14 21
18 13 20 9 4
7 2 17 22 15
12 19 24 5 10
25 6 11 16 23
1 8 3 14 21
16 13 20 9 4
7 2 15 22 19
12 17 24 5 10
25 6 11 18 23
1 14 3 8 21
4 9 20 13 16
19 2 15 22 7
10 5 24 17 12
25 18 11 6 23
1 14 3 8 21
4 9 20 13 18
15 2 17 22 7
10 5 24 19 12
25 16 11 6 23
1 12 3 18 21
4 17 20 13 8
11 2 7 22 19
16 5 24 9 14
25 10 15 6 23
1 10 3 16 21
4 15 20 11 6
9 2 5 22 17
14 19 24 7 12
25 8 13 18 23
1 22 11 16 7
12 17 8 21 10
25 2 23 6 15
18 13 4 9 20
3 24 19 14 5
1 22 11 16 7
12 17 8 21 10
23 2 25 6 15
18 13 4 9 20
3 24 19 14 5
1 24 11 16 7
12 17 8 25 10
23 2 21 6 15
18 13 4 9 20
3 22 19 14 5
1 22 11 16 7
12 17 8 23 10
21 2 19 6 15
18 13 4 9 24
3 20 25 14 5
1 20 11 16 7
12 25 8 21 10
19 2 17 6 15
24 13 4 9 22
3 18 23 14 5
1 18 11 24 7
12 23 8 19 10
17 2 15 6 25
22 13 4 9 20
3 16 21 14 5
1 16 11 22 7
12 21 8 17 10
15 2 13 6 23
20 25 4 9 18
3 14 19 24 5
1 14 25 20 7
24 19 8 15 10
13 2 11 6 21
18 23 4 9 16
3 12 17 22 5
1 24 13 18 7
14 19 8 23 12
25 2 11 6 17
20 15 4 9 22
3 10 21 16 5
1 12 15 20 7
16 21 8 25 14
11 2 13 6 19
22 17 4 9 24
3 10 23 18 5
1 12 17 22 7
18 23 8 13 16
11 2 15 6 21
24 19 4 9 14
3 10 25 20 5
1 12 19 24 7
20 25 8 13 18
11 2 17 6 23
16 21 4 9 14
3 10 15 22 5
1 12 23 18 7
24 17 8 13 22
11 2 21 6 19
16 25 4 9 14
3 10 15 20 5
1 12 25 18 7
22 17 8 13 24
11 2 23 6 19
16 21 4 9 14
3 10 15 20 5
1 12 23 18 7
22 17 8 13 24
11 2 25 6 19
16 21 4 9 14
3 10 15 20 5
1 12 21 18 7
22 17 8 13 20
11 2 19 6 25
16 23 4 9 14
3 10 15 24 5
1 16 21 10 7
20 11 8 15 22
25 2 17 6 9
12 19 4 23 14
3 24 13 18 5
1 16 21 10 7
22 11 8 15 20
17 2 25 6 9
12 23 4 19 14
3 18 13 24 5
1 16 21 10 7
22 11 8 15 20
17 2 23 6 9
12 25 4 19 14
3 18 13 24 5
1 16 25 10 7
24 11 8 15 20
17 2 21 6 9
12 23 4 19 14
3 18 13 22 5
1 16 23 10 7
22 11 8 15 24
17 2 19 6 9
12 21 4 25 14
3 18 13 20 5
1 24 19 10 7
18 11 8 25 20
23 2 15 6 9
12 17 4 21 14
3 22 13 16 5
1 22 17 10 7
16 11 8 23 18
21 2 13 6 9
12 15 4 19 24
3 20 25 14 5
1 20 15 10 7
14 25 8 21 16
19 2 11 6 9
24 13 4 17 22
3 18 23 12 5
1 18 23 12 7
24 13 8 17 22
19 2 11 6 9
14 25 4 21 16
3 20 15 10 5
1 20 25 14 7
12 15 8 19 24
21 2 13 6 9
16 11 4 23 18
3 22 17 10 5
1 22 13 16 7
12 17 8 21 14
23 2 15 6 9
18 11 4 25 20
3 24 19 10 5
1 24 13 18 7
12 19 8 23 14
25 2 17 6 9
20 11 4 15 22
3 16 21 10 5
1 18 13 20 7
12 21 8 25 14
17 2 19 6 9
22 11 4 15 24
3 16 23 10 5
1 18 13 22 7
12 23 8 19 14
17 2 21 6 9
24 11 4 15 20
3 16 25 10 5
1 18 13 24 7
12 25 8 19 14
17 2 23 6 9
22 11 4 15 20
3 16 21 10 5
1 18 13 24 7
12 23 8 19 14
17 2 25 6 9
22 11 4 15 20
3 16 21 10 5
1 24 13 18 7
14 19 8 23 12
9 2 25 6 17
20 15 4 11 22
3 10 21 16 5
1 24 13 18 7
14 19 8 25 12
9 2 23 6 17
20 15 4 11 22
3 10 21 16 5
1 22 13 18 7
14 19 8 23 12
9 2 21 6 17
20 15 4 11 24
3 10 25 16 5
1 20 13 18 7
14 25 8 21 12
9 2 19 6 17
24 15 4 11 22
3 10 23 16 5
1 18 13 24 7
14 23 8 19 12
9 2 17 6 25
22 15 4 11 20
3 10 21 16 5
1 16 13 22 7
14 21 8 17 12
9 2 15 6 23
20 25 4 11 18
3 10 19 24 5
1 14 25 20 7
24 19 8 15 12
9 2 13 6 21
18 23 4 11 16
3 10 17 22 5
1 12 23 18 7
22 17 8 13 24
9 2 11 6 19
16 21 4 25 14
3 10 15 20 5
1 10 15 20 7
16 21 8 25 14
9 2 11 6 19
22 17 4 13 24
3 12 23 18 5
1 10 17 22 7
18 23 8 11 16
9 2 13 6 21
24 19 4 15 12
3 14 25 20 5
1 10 19 24 7
20 25 8 11 18
9 2 15 6 23
14 21 4 17 12
3 16 13 22 5
1 10 23 16 7
24 15 8 11 22
9 2 19 6 17
14 25 4 21 12
3 20 13 18 5
1 10 25 16 7
20 15 8 11 24
9 2 21 6 17
14 19 4 23 12
3 22 13 18 5
1 10 21 16 7
20 15 8 11 22
9 2 23 6 17
14 19 4 25 12
3 24 13 18 5
1 10 21 16 7
20 15 8 11 22
9 2 25 6 17
14 19 4 23 12
3 24 13 18 5
1 10 21 16 7
22 15 8 11 20
9 2 17 6 25
14 23 4 19 12
3 18 13 24 5
1 18 21 12 7
20 13 8 17 22
25 2 19 6 11
14 9 4 23 16
3 24 15 10 5
1 18 23 12 7
24 13 8 17 22
19 2 25 6 11
14 9 4 21 16
3 20 15 10 5
1 18 25 12 7
24 13 8 17 22
19 2 23 6 11
14 9 4 21 16
3 20 15 10 5
1 18 23 12 7
22 13 8 17 24
19 2 21 6 11
14 9 4 25 16
3 20 15 10 5
1 24 19 12 7
18 13 8 25 20
23 2 17 6 11
14 9 4 21 16
3 22 15 10 5
1 22 17 12 7
16 13 8 23 18
21 2 15 6 11
14 9 4 19 24
3 20 25 10 5
1 20 15 12 7
14 25 8 21 16
19 2 13 6 11
24 9 4 17 22
3 18 23 10 5
1 18 13 24 7
12 23 8 19 14
17 2 11 6 25
22 9 4 15 20
3 16 21 10 5
1 20 25 14 7
10 15 8 19 24
21 2 11 6 13
16 9 4 23 18
3 22 17 12 5
1 22 11 16 7
10 17 8 21 12
23 2 13 6 15
18 9 4 25 20
3 24 19 14 5
1 24 11 18 7
10 19 8 23 12
25 2 15 6 17
20 9 4 13 22
3 14 21 16 5
1 16 11 20 7
10 21 8 25 12
15 2 17 6 19
22 9 4 13 24
3 14 23 18 5
1 16 11 22 7
10 23 8 17 12
15 2 19 6 21
24 9 4 13 18
3 14 25 20 5
1 16 11 24 7
10 25 8 17 12
15 2 21 6 23
20 9 4 13 18
3 14 19 22 5
1 16 11 22 7
10 21 8 17 12
15 2 25 6 23
20 9 4 13 18
3 14 19 24 5
1 16 11 22 7
10 21 8 17 12
15 2 23 6 25
20 9 4 13 18
3 14 19 24 5
1 20 7 14 25
10 15 24 19 8
21 2 9 6 13
16 11 4 23 18
3 22 17 12 5
1 22 7 16 25
12 17 24 21 8
23 2 11 6 15
18 13 4 9 20
3 10 19 14 5
1 12 7 18 25
22 17 24 13 8
11 2 21 6 19
16 23 4 9 14
3 10 15 20 5
1 12 7 18 25
20 17 24 13 8
11 2 19 6 23
16 21 4 9 14
3 10 15 22 5
1 18 7 12 25
8 13 24 17 20
23 2 19 6 11
14 9 4 21 16
3 22 15 10 5
1 18 7 12 25
8 13 24 17 22
19 2 21 6 11
14 9 4 23 16
3 20 15 10 5
1 16 7 22 25
8 21 24 17 12
15 2 11 6 23
20 9 4 13 18
3 14 19 10 5
1 14 7 20 25
8 19 24 15 10
13 2 9 6 21
18 23 4 11 16
3 12 17 22 5
1 16 21 8 25
22 7 24 15 20
17 2 11 6 9
12 23 4 19 14
3 18 13 10 5
1 20 13 8 25
12 7 24 19 14
21 2 15 6 9
16 11 4 23 18
3 22 17 10 5
1 22 13 8 25
12 7 24 21 14
23 2 17 6 9
18 11 4 15 20
3 16 19 10 5
1 14 19 8 25
20 7 24 13 18
15 2 9 6 23
10 21 4 17 12
3 16 11 22 5
1 12 19 14 25
18 7 24 11 20
23 2 13 6 15
8 17 4 21 10
3 22 9 16 5
1 12 17 22 25
18 7 24 11 16
13 2 21 6 23
8 19 4 15 10
3 14 9 20 5
1 12 17 20 25
18 7 24 11 16
13 2 19 6 21
8 23 4 15 10
3 14 9 22 5
1 12 21 16 25
20 7 24 11 22
13 2 15 6 17
8 19 4 23 10
3 14 9 18 5
1 16 19 10 25
18 11 24 15 20
23 2 17 6 9
12 7 4 21 14
3 22 13 8 5
1 16 21 10 25
20 11 24 15 22
17 2 19 6 9
12 7 4 23 14
3 18 13 8 5
1 16 11 22 25
10 21 24 17 12
15 2 9 6 23
20 7 4 13 18
3 14 19 8 5
1 20 9 14 25
8 15 24 19 10
21 2 11 6 13
16 7 4 23 18
3 22 17 12 5
1 22 9 16 25
8 17 24 21 10
23 2 13 6 15
18 7 4 11 20
3 12 19 14 5
1 14 9 20 25
8 19 24 15 10
13 2 21 6 23
18 7 4 11 16
3 12 17 22 5
1 10 19 14 25
18 13 24 9 20
23 2 11 6 15
12 17 4 21 8
3 22 7 16 5
1 10 15 22 25
16 21 24 9 14
11 2 19 6 23
20 17 4 13 8
3 12 7 18 5
1 10 15 20 25
16 19 24 9 14
11 2 17 6 21
18 23 4 13 8
3 12 7 22 5
1 10 21 16 25
20 15 24 9 22
11 2 13 6 17
14 19 4 23 8
3 12 7 18 5
1 16 21 10 25
22 9 24 15 20
17 2 13 6 11
8 23 4 19 14
3 18 7 12 5
1 20 15 10 25
14 9 24 19 16
21 2 17 6 11
8 13 4 23 18
3 22 7 12 5
1 22 15 10 25
14 9 24 21 16
23 2 19 6 11
8 13 4 17 20
3 18 7 12 5
1 14 19 10 25
20 9 24 13 18
15 2 11 6 23
8 21 4 17 12
3 16 7 22 5
1 12 17 6 23
16 7 22 11 18
21 2 13 24 5
8 15 4 19 10
3 20 9 14 25
1 12 17 6 23
18 7 22 11 16
13 2 19 24 5
8 21 4 15 10
3 14 9 20 25
1 12 19 6 23
18 7 22 11 20
13 2 15 24 5
8 17 4 21 10
3 14 9 16 25
1 14 19 8 23
20 9 22 13 18
15 2 7 24 5
10 21 4 17 12
3 16 11 6 25
1 18 9 12 23
8 13 22 17 10
19 2 11 24 5
14 7 4 21 16
3 20 15 6 25
1 20 9 14 23
8 15 22 19 10
21 2 13 24 5
16 7 4 11 18
3 12 17 6 25
1 14 19 10 23
20 9 22 5 18
15 2 13 24 11
8 21 4 17 6
3 16 7 12 25
1 18 15 10 23
14 9 22 5 16
19 2 17 24 11
8 13 4 21 6
3 20 7 12 25
1 20 15 10 23
14 9 22 5 16
21 2 19 24 11
8 13 4 17 6
3 18 7 12 25
1 12 17 10 23
18 9 22 5 16
13 2 11 24 21
8 19 4 15 6
3 14 7 20 25
1 6 17 12 23
16 11 22 5 18
21 2 7 24 13
10 15 4 19 8
3 20 9 14 25
1 6 11 20 23
12 19 22 5 10
7 2 15 24 21
18 13 4 9 16
3 8 17 14 25
1 6 11 18 23
12 17 22 5 10
7 2 13 24 19
16 21 4 9 14
3 8 15 20 25
1 6 19 14 23
18 13 22 5 20
7 2 9 24 15
12 17 4 21 10
3 8 11 16 25
1 14 19 6 23
20 5 22 13 18
15 2 9 24 7
10 21 4 17 12
3 16 11 8 25
1 18 11 6 23
10 5 22 17 12
19 2 13 24 7
14 9 4 21 16
3 20 15 8 25
1 20 11 6 23
10 5 22 19 12
21 2 15 24 7
16 9 4 13 18
3 14 17 8 25
1 12 17 6 23
18 5 22 11 16
13 2 7 24 21
8 19 4 15 10
3 14 9 20 25
1 10 17 12 23
16 5 22 9 18
21 2 11 24 13
6 15 4 19 8
3 20 7 14 25
1 10 15 20 23
16 5 22 9 14
11 2 19 24 21
6 17 4 13 8
3 12 7 18 25
1 10 15 18 23
16 5 22 9 14
11 2 17 24 19
6 21 4 13 8
3 12 7 20 25
1 10 19 14 23
18 5 22 9 20
11 2 13 24 15
6 17 4 21 8
3 12 7 16 25
1 14 9 20 23
10 19 22 15 8
5 2 13 24 21
18 11 4 7 16
3 6 17 12 25
1 12 9 18 23
10 17 22 13 8
5 2 11 24 19
16 21 4 7 14
3 6 15 20 25
1 8 19 14 23
18 13 22 9 20
5 2 7 24 15
12 17 4 21 10
3 6 11 16 25
1 6 19 12 23
20 11 22 7 18
5 2 15 24 13
10 21 4 17 8
3 16 9 14 25
1 6 17 12 23
16 11 22 7 18
5 2 19 24 13
10 15 4 21 8
3 20 9 14 25
1 6 17 12 23
18 11 22 7 16
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