Probability of Knight to remain in the chessboard

Given an NxN chessboard and a Knight at position (x,y). The Knight has to take exactly K steps, where at each step it chooses any of the 8 directions uniformly at random. What is the probability that the Knight remains in the chessboard after taking K steps, with the condition that it can’t enter the board again once it leaves it?
Examples:

Let's take:
8x8 chessboard,
initial position of the knight : (0, 0),
number of steps : 1
At each step, the Knight has 8 different positions to choose from. 

If it starts from (0, 0), after taking one step it will lie inside the
board only at 2 out of 8 positions, and will lie outside at other positions.
So, the probability is 2/8 = 0.25

Approach:

One thing that we can observe is that at every step the Knight has 8 choices to choose from. Suppose, the Knight has to take k steps and after taking the Kth step the knight reaches (x,y). There are 8 different positions from where the Knight can reach to (x,y) in one step, and they are: (x+1,y+2), (x+2,y+1), (x+2,y-1), (x+1,y-2), (x-1,y-2), (x-2,y-1), (x-2,y+1), (x-1,y+2). 
What if we already knew the probabilities of reaching these 8 positions after K-1 steps? 

Then, the final probability after K steps will simply be equal to the (? probability of reaching each of these 8 positions after K-1 steps)/8; 

Here we are dividing by 8 because each of these 8 positions has 8 choices and position (x,y) is one of the choices. 

For the positions that lie outside the board, we will either take their probabilities as 0 or simply neglect it.

Since we need to keep track of the probabilities at each position for every number of steps, we need Dynamic Programming to solve this problem. 
We are going to take an array dp[x][y][steps] which will store the probability of reaching (x,y) after (steps) number of moves. 

Base case: if the number of steps is 0, then the probability that the Knight will remain inside the board is 1.
Below is the implementation of the above approach: 

C++

// C++ program to find the probability of the
// Knight to remain inside the chessboard after
// taking exactly K number of steps
#include <bits/stdc++.h>
using namespace std;

// size of the chessboard
#define N 8

// direction vector for the Knight
int dx[] = { 1, 2, 2, 1, -1, -2, -2, -1 };
int dy[] = { 2, 1, -1, -2, -2, -1, 1, 2 };

// returns true if the knight is inside the chessboard
bool inside(int x, int y)
{
    return (x >= 0 and x < N and y >= 0 and y < N);
}

// Bottom up approach for finding the probability to
// go out of chessboard.
double findProb(int start_x, int start_y, int steps)
{
    // dp array
    double dp1[N][N][steps + 1];

    // for 0 number of steps, each position
    // will have probability 1
    for (int i = 0; i < N; ++i)
        for (int j = 0; j < N; ++j)
            dp1[i][j][0] = 1;

    // for every number of steps s
    for (int s = 1; s <= steps; ++s) {

        // for every position (x,y) after
        // s number of steps
        for (int x = 0; x < N; ++x) {
            for (int y = 0; y < N; ++y) {
                double prob = 0.0;

                // for every position reachable from (x,y)
                for (int i = 0; i < 8; ++i) {
                    int nx = x + dx[i];
                    int ny = y + dy[i];

                    // if this position lie inside the board
                    if (inside(nx, ny))
                        prob += dp1[nx][ny][s - 1] / 8.0;
                }

                // store the result
                dp1[x][y][s] = prob;
            }
        }
    }

    // return the result
    return dp1[start_x][start_y][steps];
}

// Driver Code
int main()
{
    // number of steps
    int K = 3;

    // Function Call
    cout << findProb(0, 0, K) << endl;

    return 0;
}

Java

// Java program to find the probability
// of the Knight to remain inside the
// chessboard after taking exactly K
// number of steps
class GFG {

    // size of the chessboard
    static final int N = 8;

    // direction vector for the Knight
    static int dx[] = { 1, 2, 2, 1, -1, -2, -2, -1 };

    static int dy[] = { 2, 1, -1, -2, -2, -1, 1, 2 };

    // returns true if the knight is
    // inside the chessboard
    static boolean inside(int x, int y)
    {
        return (x >= 0 && x < N && y >= 0 && y < N);
    }

    // Bottom up approach for finding
    // the probability to go out of
    // chessboard.
    static double findProb(int start_x, int start_y,
                           int steps)
    {

        // dp array
        double dp1[][][] = new double[N][N][steps + 1];

        // for 0 number of steps, each position
        // will have probability 1
        for (int i = 0; i < N; ++i)
            for (int j = 0; j < N; ++j)
                dp1[i][j][0] = 1;

        // for every number of steps s
        for (int s = 1; s <= steps; ++s) {

            // for every position (x, y) after
            // s number of steps
            for (int x = 0; x < N; ++x) {

                for (int y = 0; y < N; ++y) {

                    double prob = 0.0;

                    // for every position reachable
                    // from (x, y)
                    for (int i = 0; i < 8; ++i) {
                        int nx = x + dx[i];
                        int ny = y + dy[i];

                        // if this position lie
                        // inside the board
                        if (inside(nx, ny))
                            prob
                                += dp1[nx][ny][s - 1] / 8.0;
                    }

                    // store the result
                    dp1[x][y][s] = prob;
                }
            }
        }

        // return the result
        return dp1[start_x][start_y][steps];
    }

    // Driver code
    public static void main(String[] args)
    {

        // number of steps
        int K = 3;

        // Function Call
        System.out.println(findProb(0, 0, K));
    }
}

// This code is contributed by Anant Agarwal.

Python3

# Python3 program to find the probability of
# the Knight to remain inside the chessboard
# after taking exactly K number of steps
# size of the chessboard
N = 8

# Direction vector for the Knight
dx = [1, 2, 2, 1, -1, -2, -2, -1]
dy = [2, 1, -1, -2, -2, -1, 1, 2]

# returns true if the knight
# is inside the chessboard

def inside(x, y):
    return (x >= 0 and x < N and y >= 0 and y < N)

# Bottom up approach for finding the
# probability to go out of chessboard.

def findProb(start_x, start_y, steps):

    # dp array
    dp1 = [[[0 for i in range(N+5)]
            for j in range(N+5)]
            for k in range(steps + 5)]

    # For 0 number of steps, each
    # position will have probability 1
    for i in range(N):
        for j in range(N):
            dp1[i][j][0] = 1

    # for every number of steps s
    for s in range(1, steps + 1):

        # for every position (x,y) after
        # s number of steps
        for x in range(N):

            for y in range(N):
                prob = 0.0

                # For every position reachable from (x,y)
                for i in range(8):
                    nx = x + dx[i]
                    ny = y + dy[i]

                    # if this position lie inside the board
                    if (inside(nx, ny)):
                        prob += dp1[nx][ny][s-1] / 8.0

                # store the result
                dp1[x][y][s] = prob

    # return the result
    return dp1[start_x][start_y][steps]

# Driver code

# number of steps
K = 3

# Function Call
print(findProb(0, 0, K))

# This code is contributed by Anant Agarwal.

C#

// C# program to find the
// probability of the Knight
// to remain inside the
// chessboard after taking
// exactly K number of steps
using System;

class GFG {

    // size of the chessboard
    static int N = 8;

    // direction vector
    // for the Knight
    static int[] dx = { 1, 2, 2, 1, -1, -2, -2, -1 };

    static int[] dy = { 2, 1, -1, -2, -2, -1, 1, 2 };

    // returns true if the
    // knight is inside the
    // chessboard
    static bool inside(int x, int y)
    {
        return (x >= 0 && x < N && y >= 0 && y < N);
    }

    // Bottom up approach for
    // finding the probability
    // to go out of chessboard.
    static double findProb(int start_x, int start_y,
                           int steps)
    {

        // dp array
        double[, , ] dp1 = new double[N, N, steps+1];

        // for 0 number of steps,
        // each position will have
        // probability 1
        for (int i = 0; i < N; ++i)
            for (int j = 0; j < N; ++j)
                dp1[i, j, 0] = 1;

        // for every number
        // of steps s
        for (int s = 1; s <= steps; ++s) {

            // for every position (x, y)
            // after s number of steps
            for (int x = 0; x < N; ++x) {
                for (int y = 0; y < N; ++y) {
                    double prob = 0.0;

                    // for every position
                    // reachable from (x, y)
                    for (int i = 0; i < 8; ++i) {
                        int nx = x + dx[i];
                        int ny = y + dy[i];

                        // if this position lie
                        // inside the board
                        if (inside(nx, ny))
                            prob
                                += dp1[nx, ny, s - 1] / 8.0;
                    }

                    // store the result
                    dp1[x, y, s] = prob;
                }
            }
        }

        // return the result
        return dp1[start_x, start_y, steps];
    }

    // Driver code
    static void Main()
    {
        // number of steps
        int K = 3;

        // Function Call
        Console.WriteLine(findProb(0, 0, K));
    }
}

// This code is contributed
// by Sam007

PHP

<?php
// PHP program to find the probability
// of the Knight to remain inside the
// chessboard after taking exactly K
// number of steps

// size of the chessboard
$N = 8;

// direction vector for the Knight
$dx = array(1, 2, 2, 1, -1, -2, -2, -1 );
$dy = array(2, 1, -1, -2, -2, -1, 1, 2 );

// returns true if the knight is
// inside the chessboard
function inside($x, $y)
{
    global $N;
    return ($x >= 0 and $x < $N and
            $y >= 0 and $y < $N);
}

// Bottom up approach for finding the
// probability to go out of chessboard.
function findProb($start_x, $start_y, $steps)
{
    global $N, $dx, $dy;

    // dp array
    $dp1 = array_fill(0, $N,
           array_fill(0, $N,
           array_fill(0, $steps+1, NULL)));

    // for 0 number of steps, each
    // position will have probability 1
    for ($i = 0; $i < $N; ++$i)
        for ($j = 0; $j < $N; ++$j)
            $dp1[$i][$j][0] = 1;

    // for every number of steps s
    for ($s = 1; $s <= $steps; ++$s)
    {
        // for every position (x,y) after
        // s number of steps
        for ($x = 0; $x < $N; ++$x)
        {
            for ($y = 0; $y < $N; ++$y)
            {
                $prob = 0.0;

                // for every position
                // reachable from (x,y)
                for ($i = 0; $i < 8; ++$i)
                {
                    $nx = $x + $dx[$i];
                    $ny = $y + $dy[$i];

                    // if this position lie inside
                    // the board
                    if (inside($nx, $ny))
                        $prob += $dp1[$nx][$ny][$s - 1] / 8.0;
                }

                // store the result
                $dp1[$x][$y][$s] = $prob;
            }
        }
    }

    // return the result
    return $dp1[$start_x][$start_y][$steps];
}

// Driver Code

// number of steps
$K = 3;

// Function Call
echo findProb(0, 0, $K) . "\n";

// This code is contributed by ita_c
?>

Javascript

<script>

// Javascript program to find the probability
// of the Knight to remain inside the
// chessboard after taking exactly K
// number of steps

    // size of the chessboard
    let N = 8;

    // direction vector for the Knight
    let dx = [ 1, 2, 2, 1, -1, -2, -2, -1 ];

    let dy = [2, 1, -1, -2, -2, -1, 1, 2];

    // returns true if the knight is
    // inside the chessboard
    function inside(x,y)
    {
        return (x >= 0 && x < N && y >= 0 && y < N);
    }

    // Bottom up approach for finding
    // the probability to go out of
    // chessboard.
    function findProb(start_x, start_y, steps)
    {
        // dp array
        let dp1 = new Array(N);
        for(let i = 0; i < N; i++)
        {
            dp1[i] = new Array(N);
            for(let j = 0; j < N; j++)
            {
                dp1[i][j] = new Array(steps + 1);
                for(let k = 0; k < steps + 1; k++)
                {
                    dp1[i][j][k] = 0;
                }
            }
        }

        // for 0 number of steps, each position
        // will have probability 1
        for (let i = 0; i < N; ++i)
            for (let j = 0; j < N; ++j)
                dp1[i][j][0] = 1;

        // for every number of steps s
        for (let s = 1; s <= steps; ++s)
        {

            // for every position (x, y) after
            // s number of steps
            for (let x = 0; x < N; ++x)
            {

                for (let y = 0; y < N; ++y)
                {

                    let prob = 0.0;

                    // for every position reachable
                    // from (x, y)
                    for (let i = 0; i < 8; ++i)
                    {
                        let nx = x + dx[i];
                        let ny = y + dy[i];

                        // if this position lie
                        // inside the board
                        if (inside(nx, ny))
                            prob
                                += dp1[nx][ny][s - 1] / 8.0;
                    }

                    // store the result
                    dp1[x][y][s] = prob;
                }
            }
        }

        // return the result
        return dp1[start_x][start_y][steps];
    }

     // Driver code

    // number of steps
    let K = 3;

    // Function Call
    document.write(findProb(0, 0, K));

    // This code is contributed by rag2127
</script>

Output

0.125

Time Complexity: O(NxNxKx8) which is O(NxNxK), where N is the size of the board and K is the number of steps. 
Space Complexity: O(NxNxK)

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