Find the integer points (x, y) with Manhattan distance atleast N

Given a number N, the task is to find the integer points (x, y) such that 0 <= x, y <= N and Manhattan distance between any two points will be atleast N.
Examples:

Input: N = 3
Output: (0, 0) (0, 3) (3, 0) (3, 3)

Input: N = 4
Output: (0, 0) (0, 4) (4, 0) (4, 4) (2, 2)

Approach:

Manhattan Distance between two points (x₁, y₁) and (x₂, y₂) is:
|x₁ – x₂| + |y₁ – y₂|
Here for all pair of points this distance will be atleast N.
As 0 <= x <= N and 0 <= y <= N so we can imagine a square of side length N whose bottom left corner is (0, 0) and top right corner is (N, N).
So if we place 4 points in this corner then Manhattan distance will be atleast N.
Now as we have to maximize the number of the point we have to check is there any available point inside the square.
If N is even then middle point of the square which is (N/2, N/2) is integer point, otherwise, it will be float value as N/2 is not a integer when N is odd.
So the only available position is the middle point and we can put a point there only if N is even.
So number of points will be 4 if N is odd and if N is even then the number of points will be 5.

Below is the implementation of the above approach:

C++

// C++ code to Find the integer points (x, y)
// with Manhattan distance atleast N
 
#include <bits/stdc++.h>
using namespace std;
 
// C++ function to find all possible point
vector<pair<int, int> > FindPoints(int n)
{
 
    vector<pair<int, int> > v;
 
    // Find all 4 corners of the square
    // whose side length is n
    v.push_back({ 0, 0 });
    v.push_back({ 0, n });
    v.push_back({ n, 0 });
    v.push_back({ n, n });
 
    // If n is even then the middle point
    // of the square will be an integer,
    // so we will take that point
    if (n % 2 == 0)
        v.push_back({ n / 2, n / 2 });
 
    return v;
}
 
// Driver Code
int main()
{
 
    int N = 8;
 
    vector<pair<int, int> > v
        = FindPoints(N);
 
    // Printing all possible points
    for (auto i : v) {
        cout << "(" << i.first << ", "
             << i.second << ") ";
    }
    return 0;
}

Java

// Java code to Find the integer points (x, y)
// with Manhattan distance atleast N
import java.util.*;
 
class GFG
{
 
static class pair
{
    int first, second;
    public pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}
 
// Java function to find all possible point
static Vector<pair> FindPoints(int n)
{
    Vector<pair> v = new Vector<pair>();
 
    // Find all 4 corners of the square
    // whose side length is n
    v.add(new pair( 0, 0 ));
    v.add(new pair( 0, n ));
    v.add(new pair( n, 0 ));
    v.add(new pair( n, n ));
 
    // If n is even then the middle point
    // of the square will be an integer,
    // so we will take that point
    if (n % 2 == 0)
        v.add(new pair( n / 2, n / 2 ));
 
    return v;
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 8;
 
    Vector<pair > v = FindPoints(N);
 
    // Printing all possible points
    for (pair i : v)
    {
        System.out.print("(" + i.first + ", " +
                               i.second + ") ");
    }
}
}
 
// This code is contributed by PrinciRaj1992

Python3

# Python3 code to Find the integer points (x, y)
# with Manhattan distance atleast N
 
# function to find all possible point
def FindPoints(n) :
 
    v = [];
 
    # Find all 4 corners of the square
    # whose side length is n
    v.append([ 0, 0 ]);
    v.append([ 0, n ]);
    v.append([ n, 0 ]);
    v.append([ n, n ]);
 
    # If n is even then the middle point
    # of the square will be an integer,
    # so we will take that point
    if (n % 2 == 0) :
        v.append([ n // 2, n // 2 ]);
 
    return v;
 
# Driver Code
if __name__ == "__main__" :
 
    N = 8;
 
    v = FindPoints(N);
 
    # Printing all possible points
    for element in v :
        print("(", element[0],
              ",", element[1], ")", end = " ");
 
# This code is contributed by AnkitRai01

C#

// C# code to Find the integer points (x, y)
// with Manhattan distance atleast N
using System;
using System.Collections.Generic;
 
class GFG
{
 
class pair
{
    public int first, second;
    public pair(int first, int second)
    {
        this.first = first;
        this.second = second;
    }
}
 
// Function to find all possible point
static List<pair> FindPoints(int n)
{
    List<pair> v = new List<pair>();
 
    // Find all 4 corners of the square
    // whose side length is n
    v.Add(new pair( 0, 0 ));
    v.Add(new pair( 0, n ));
    v.Add(new pair( n, 0 ));
    v.Add(new pair( n, n ));
 
    // If n is even then the middle point
    // of the square will be an integer,
    // so we will take that point
    if (n % 2 == 0)
        v.Add(new pair( n / 2, n / 2 ));
 
    return v;
}
 
// Driver Code
public static void Main(String[] args)
{
    int N = 8;
 
    List<pair > v = FindPoints(N);
 
    // Printing all possible points
    foreach (pair i in v)
    {
        Console.Write("(" + i.first + ", " +
                            i.second + ") ");
    }
}
}
 
// This code is contributed by Rajput-Ji

Javascript

<script>
 
// Javascript code to Find the integer points (x, y)
// with Manhattan distance atleast N
 
// C++ function to find all possible point
function FindPoints(n)
{
 
    var v = [];
 
    // Find all 4 corners of the square
    // whose side length is n
    v.push([ 0, 0 ]);
    v.push([ 0, n ]);
    v.push([ n, 0 ]);
    v.push([ n, n ]);
 
    // If n is even then the middle point
    // of the square will be an integer,
    // so we will take that point
    if (n % 2 == 0)
        v.push([ n / 2, n / 2 ]);
 
    return v;
}
 
// Driver Code
var N = 8;
var v = FindPoints(N);
// Printing all possible points
v.forEach(i => {
    document.write( "(" + i[0] + ", "
         + i[1] + ") ");
});
 
// This code is contributed by rrrtnx.
</script>