Count all possible position that can be reached by Modified Knight

Given a chessboard of size 8 x 8 and the current position of Mirandote. All the rules of this chess game are the same but the knight is modified. We call the new knight “Mirandote”. The move of Mirandote is given by a blue color where its current position is denoted by red color in the following image :

The task is to find how many possible positions exist in Chessboard that can be reached by Mirandote in exactly S steps.

Examples:

Input: row = 4, col = 4, steps = 1 
Output: 12 
All the 12 moves denoted by the following image by blue color : 

Input: row = 4, col = 4, steps = 2 
Output: 55 

Solution:
We can observe that all the possible positions with respect to the current position can be written in the form of rows and columns. This is illustrated by the following image: 

We can call a function recursively for each possible position and count all the possible positions.

Below is the required implementation to find the positions:

C++

// C++ implementation to find the
// possible positions
#include <bits/stdc++.h>
using namespace std;

// Function to find the positions
void findSteps(int current_row, int current_column,
               int curr, int board_size, int steps,
               int* visited)
{
    // Bound checking
    if (current_row >= board_size || current_row < 0
        || current_column >= board_size || current_column < 0
        || curr > steps) {
        return;
    }

    // If steps is equal to current steps,
    // that means current position is reached by Mirandote
    if (curr == steps) {
        *((visited + (current_row)*board_size) + current_column) = 1;
        return;
    }

    // Recursive calls for each possible position.
    // Position of a, b, c, ..., l given in above image.
    /* a = */ findSteps(current_row - 2, current_column - 1,
                        curr + 1, board_size, steps, visited);

    /* b = */ findSteps(current_row - 2, current_column + 1,
                        curr + 1, board_size, steps, visited);

    /* c = */ findSteps(current_row - 1, current_column - 2,
                        curr + 1, board_size, steps, visited);

    /* d = */ findSteps(current_row - 1, current_column - 1,
                        curr + 1, board_size, steps, visited);

    /* e = */ findSteps(current_row - 1, current_column + 1,
                        curr + 1, board_size, steps, visited);

    /* f = */ findSteps(current_row - 1, current_column + 2,
                        curr + 1, board_size, steps, visited);

    /* g = */ findSteps(current_row + 1, current_column - 2,
                        curr + 1, board_size, steps, visited);

    /* h = */ findSteps(current_row + 1, current_column - 1,
                        curr + 1, board_size, steps, visited);

    /* i = */ findSteps(current_row + 1, current_column + 1,
                        curr + 1, board_size, steps, visited);

    /* j = */ findSteps(current_row + 1, current_column + 2,
                        curr + 1, board_size, steps, visited);

    /* k = */ findSteps(current_row + 2, current_column - 1,
                        curr + 1, board_size, steps, visited);

    /* l = */ findSteps(current_row + 2, current_column + 1,
                        curr + 1, board_size, steps, visited);

    return;
}

int countSteps(int current_row, int current_column,
               int board_size, int steps)
{

    // Visited array
    int visited[board_size][board_size];

    // Initialize visited array to zero
    for (int i = 0; i < board_size; i++) {
        for (int j = 0; j < board_size; j++) {
            visited[i][j] = 0;
        }
    }

    int answer = 0;

    // Function call where initial step count is 0
    findSteps(current_row, current_column, 0,
              board_size, steps, (int*)visited);

    for (int i = 0; i < board_size; i++) {
        for (int j = 0; j < board_size; j++) {

            // If value of element is 1, that implies,
            // the position can be reached by Mirandote.
            if (visited[i][j] == 1) {
                answer++;
            }
        }
    }

    return answer;
}

// Driver code
int main()
{
    int board_size = 8, steps = 1;
    int current_row = 4, current_column = 4;

    cout << countSteps(current_row, current_column,
                       board_size, steps);
    return 0;
}

Java

// Java implementation to find the
// possible positions
import java.util.*;

class GFG{

static int [][] visited = new int [500][500];

// Function to find the positions
static void findSteps(int current_row,
                      int current_column,
                      int curr, int board_size,
                      int steps)
{

    // Bound checking
    if (current_row >= board_size ||
        current_row < 0 ||
        current_column >= board_size ||
        current_column < 0 || curr > steps)
    {
        return;
    }

    // If steps is equal to current steps,
    // that means current position is
    // reached by Mirandote
    if (curr == steps)
    {
        visited[current_row][current_column] = 1;
        return;
    }

    // Recursive calls for each possible position.
    // Position of a, b, c, ..., l given in
    // above image.
    /* a = */ findSteps(current_row - 2,
                     current_column - 1,
                               curr + 1,
                     board_size, steps);

    /* b = */ findSteps(current_row - 2,
                     current_column + 1,
                               curr + 1,
                     board_size, steps);

    /* c = */ findSteps(current_row - 1,
                     current_column - 2,
                               curr + 1,
                     board_size, steps);

    /* d = */ findSteps(current_row - 1,
                     current_column - 1,
                               curr + 1,
                     board_size, steps);

    /* e = */ findSteps(current_row - 1,
                     current_column + 1,
                               curr + 1,
                     board_size, steps);

    /* f = */ findSteps(current_row - 1,
                     current_column + 2,
                               curr + 1,
                     board_size, steps);

    /* g = */ findSteps(current_row + 1,
                     current_column - 2,
                               curr + 1,
                     board_size, steps);

    /* h = */ findSteps(current_row + 1,
                     current_column - 1,
                               curr + 1,
                     board_size, steps);

    /* i = */ findSteps(current_row + 1,
                     current_column + 1,
                               curr + 1,
                     board_size, steps);

    /* j = */ findSteps(current_row + 1,
                     current_column + 2,
                               curr + 1,
                     board_size, steps);

    /* k = */ findSteps(current_row + 2,
                     current_column - 1,
                               curr + 1,
                     board_size, steps);

    /* l = */ findSteps(current_row + 2,
                     current_column + 1,
                               curr + 1,
                     board_size, steps);
}

static int countSteps(int current_row,
                      int current_column,
                      int board_size, int steps)
{

    // Initialize visited array to zero
    for(int i = 0; i < board_size; i++)
    {
        for(int j = 0; j < board_size; j++)
        {
            visited[i][j] = 0;
        }
    }

    int answer = 0;

    // Function call where initial step count is 0
    findSteps(current_row, current_column, 0,
              board_size,steps);

    for(int i = 0; i < board_size; i++)
    {
        for(int j = 0; j < board_size; j++)
        {

            // If value of element is 1, that implies,
            // the position can be reached by Mirandote.
            if (visited[i][j] == 1)
            {
                answer++;
            }
        }
    }
    return answer;
}

// Driver code
public static void main(String[] args)
{
    int board_size = 8, steps = 1;
    int current_row = 4, current_column = 4;

    System.out.print(countSteps(current_row,
                                current_column,
                                board_size, steps));
}
}

// This code is contributed by Stream_Cipher

Python3

# Python3 implementation to find the possible positions
visited = [[0 for i in range(500)] for j in range(500)]

# Function to find the positions
def findSteps(current_row, current_column, curr, board_size, steps):
    global visited
    # Bound checking
    if current_row >= board_size or current_row < 0 or current_column >= board_size or current_column < 0 or curr > steps:
        return

    # If steps is equal to current steps,
    # that means current position is
    # reached by Mirandote
    if curr == steps:
        visited[current_row][current_column] = 1
        return

    # Recursive calls for each possible position.
    # Position of a, b, c, ..., l given in
    # above image.
    """ a = """
    findSteps(current_row - 2, current_column - 1, curr + 1, board_size, steps)

    """ b = """
    findSteps(current_row - 2, current_column + 1, curr + 1, board_size, steps)

    """ c = """
    findSteps(current_row - 1, current_column - 2, curr + 1, board_size, steps)

    """ d = """
    findSteps(current_row - 1, current_column - 1, curr + 1, board_size, steps)

    """ e = """
    findSteps(current_row - 1, current_column + 1, curr + 1, board_size, steps)

    """ f = """
    findSteps(current_row - 1, current_column + 2, curr + 1, board_size, steps)

    """ g = """
    findSteps(current_row + 1, current_column - 2, curr + 1, board_size, steps)

    """ h = """
    findSteps(current_row + 1, current_column - 1, curr + 1, board_size, steps)

    """ i = """
    findSteps(current_row + 1, current_column + 1, curr + 1, board_size, steps)

    """ j = """
    findSteps(current_row + 1, current_column + 2, curr + 1, board_size, steps)

    """ k = """
    findSteps(current_row + 2, current_column - 1, curr + 1, board_size, steps)

    """ l = """
    findSteps(current_row + 2, current_column + 1, curr + 1, board_size, steps)

def countSteps(current_row, current_column, board_size, steps):

    # Initialize visited array to zero
    for i in range(board_size):
        for j in range(board_size):
            visited[i][j] = 0

    answer = 0

    # Function call where initial step count is 0
    findSteps(current_row, current_column, 0, board_size,steps)

    for i in range(board_size):
        for j in range(board_size):
            # If value of element is 1, that implies,
            # the position can be reached by Mirandote.
            if visited[i][j] == 1:
                answer+=1
    return answer

board_size, steps = 8, 1
current_row, current_column = 4, 4

print(countSteps(current_row, current_column, board_size, steps))

# This code is contributed by rameshtravel07.

C#

// C# implementation to find the
// possible positions
using System.Collections.Generic;
using System;

class GFG{

static int [,] visited = new int[500, 500];

// Function to find the positions
static void findSteps(int current_row,
                      int current_column,
                      int curr, int board_size,
                      int steps)
{

    // Bound checking
    if (current_row >= board_size ||
        current_row < 0  ||
        current_column >= board_size ||
        current_column < 0 || curr > steps)
    {
        return;
    }

    // If steps is equal to current steps,
    // that means current position is
    // reached by Mirandote
    if (curr == steps)
    {
        visited[current_row, current_column] = 1;
        return;
    }

    // Recursive calls for each possible position.
    // Position of a, b, c, ..., l given in above image.
    /* a = */ findSteps(current_row - 2,
                     current_column - 1,
                               curr + 1,
                     board_size, steps);

    /* b = */ findSteps(current_row - 2,
                     current_column + 1,
                               curr + 1,
                     board_size, steps);

    /* c = */ findSteps(current_row - 1,
                     current_column - 2,
                               curr + 1,
                     board_size, steps);

    /* d = */ findSteps(current_row - 1,
                     current_column - 1,
                               curr + 1,
                     board_size, steps);

    /* e = */ findSteps(current_row - 1,
                     current_column + 1,
                               curr + 1,
                     board_size, steps);

    /* f = */ findSteps(current_row - 1,
                     current_column + 2,
                               curr + 1,
                     board_size, steps);

    /* g = */ findSteps(current_row + 1,
                     current_column - 2,
                               curr + 1,
                     board_size, steps);

    /* h = */ findSteps(current_row + 1,
                     current_column - 1,
                               curr + 1,
                     board_size, steps);

    /* i = */ findSteps(current_row + 1,
                     current_column + 1,
                               curr + 1,
                     board_size, steps);

    /* j = */ findSteps(current_row + 1,
                     current_column + 2,
                               curr + 1,
                     board_size, steps);

    /* k = */ findSteps(current_row + 2,
                     current_column - 1,
                               curr + 1,
                     board_size, steps);

    /* l = */ findSteps(current_row + 2,
                     current_column + 1,
                               curr + 1,
                     board_size, steps);
}

static int countSteps(int current_row,
                      int current_column,
                      int board_size, int steps)
{

    // Initialize visited array to zero
    for(int i = 0; i < board_size; i++)
    {
        for(int j = 0; j < board_size; j++)
        {
            visited[i, j] = 0;
        }
    }

    int answer = 0;

    // Function call where initial step count is 0
    findSteps(current_row, current_column, 0,
              board_size,steps);

    for(int i = 0; i < board_size; i++)
    {
        for(int j = 0; j < board_size; j++)
        {

            // If value of element is 1,
            // that implies, the position
            // can be reached by Mirandote.
            if (visited[i, j] == 1)
            {
                answer++;
            }
        }
    }
    return answer;
}

// Driver code
public static void Main()
{
    int board_size = 8, steps = 1;
    int current_row = 4, current_column = 4;

    Console.WriteLine(countSteps(current_row,
                                 current_column,
                                 board_size, steps));
}
}

// This code is contributed by Stream_Cipher

Javascript

<script>
    // Javascript implementation to find the
    // possible positions

    let visited = new Array(500);

    // Function to find the positions
    function findSteps(current_row, current_column, curr, board_size, steps)
    {

        // Bound checking
        if (current_row >= board_size ||
            current_row < 0 ||
            current_column >= board_size ||
            current_column < 0 || curr > steps)
        {
            return;
        }

        // If steps is equal to current steps,
        // that means current position is
        // reached by Mirandote
        if (curr == steps)
        {
            visited[current_row][current_column] = 1;
            return;
        }

        // Recursive calls for each possible position.
        // Position of a, b, c, ..., l given in
        // above image.
        /* a = */ findSteps(current_row - 2,
                         current_column - 1,
                                   curr + 1,
                         board_size, steps);

        /* b = */ findSteps(current_row - 2,
                         current_column + 1,
                                   curr + 1,
                         board_size, steps);

        /* c = */ findSteps(current_row - 1,
                         current_column - 2,
                                   curr + 1,
                         board_size, steps);

        /* d = */ findSteps(current_row - 1,
                         current_column - 1,
                                   curr + 1,
                         board_size, steps);

        /* e = */ findSteps(current_row - 1,
                         current_column + 1,
                                   curr + 1,
                         board_size, steps);

        /* f = */ findSteps(current_row - 1,
                         current_column + 2,
                                   curr + 1,
                         board_size, steps);

        /* g = */ findSteps(current_row + 1,
                         current_column - 2,
                                   curr + 1,
                         board_size, steps);

        /* h = */ findSteps(current_row + 1,
                         current_column - 1,
                                   curr + 1,
                         board_size, steps);

        /* i = */ findSteps(current_row + 1,
                         current_column + 1,
                                   curr + 1,
                         board_size, steps);

        /* j = */ findSteps(current_row + 1,
                         current_column + 2,
                                   curr + 1,
                         board_size, steps);

        /* k = */ findSteps(current_row + 2,
                         current_column - 1,
                                   curr + 1,
                         board_size, steps);

        /* l = */ findSteps(current_row + 2,
                         current_column + 1,
                                   curr + 1,
                         board_size, steps);
    }

    function countSteps(current_row, current_column, board_size, steps)
    {

        // Initialize visited array to zero
        for(let i = 0; i < board_size; i++)
        {
            visited[i] = new Array(board_size);
            for(let j = 0; j < board_size; j++)
            {
                visited[i][j] = 0;
            }
        }

        let answer = 0;

        // Function call where initial step count is 0
        findSteps(current_row, current_column, 0,
                  board_size,steps);

        for(let i = 0; i < board_size; i++)
        {
            for(let j = 0; j < board_size; j++)
            {

                // If value of element is 1, that implies,
                // the position can be reached by Mirandote.
                if (visited[i][j] == 1)
                {
                    answer++;
                }
            }
        }
        return answer;
    }

    let board_size = 8, steps = 1;
    let current_row = 4, current_column = 4;

    document.write(countSteps(current_row,
                                current_column,
                                board_size, steps));

</script>

Output:

12

Time complexity of the above algorithm is O(12^S), where S is the number of steps.

Space Complexity: O(n) where n is recursion stack space.