# Maximum integral co-ordinates with non-integer distances

Given a maximum limit of x – coordinate and y – coordinate, we want to calculate a set of coordinates such that the distance between any two points is a non-integer number. The coordinates (i, j) chosen should be of range 0<=i<=x and 0<=j<=y. Also, we have to maximize the set. Examples:

Input : 4 4 Output : 0 4 1 3 2 2 3 1 4 0 Explanation : Distance between any two points mentioned in output is not integer.

Firstly, we want to create a set, that means our set cannot contain any other point with same x’s or y’s which are used before. Well, the reason behind it is that such points which either have same x-coordinate or y-coordinate would cancel that coordinate, resulting an integral distance between them.

Example, consider points (1, 4) and (1, 5), the x-coordinate would cancel and thus, we will get and integral distance.

Secondly, we can observe that, we have only x+1 distinct i-coordinates and y+1 distinct j-coordinates. Thus, the size of the set cannot exceed min(x, y)+1.

Third observation is that we know that the diagonal elements are |i-j|* distance apart, thus, we take evaluate along the diagonal element of i-coordinate and calculate the j-coordinate by formula min(i, j)-i.

`// C++ program to find maximum integral points ` `// such that distances between any two is not ` `// integer. ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Making set of coordinates such that ` `// any two points are non-integral distance apart ` `void` `printSet(` `int` `x, ` `int` `y) ` `{ ` ` ` `// used to avoid duplicates in result ` ` ` `set<pair<` `int` `, ` `int` `> > arr; ` ` ` ` ` `for` `(` `int` `i = 0; i <= min(x, y); i++) { ` ` ` ` ` `pair<` `int` `, ` `int` `> pq; ` ` ` `pq = make_pair(i, min(x, y) - i); ` ` ` `arr.insert(pq); ` ` ` `} ` ` ` ` ` `for` `(` `auto` `it = arr.begin(); it != arr.end(); it++) ` ` ` `cout << (*it).first << ` `" "` `<< (*it).second << endl; ` `} ` ` ` `// Driver function ` `int` `main() ` `{ ` ` ` `int` `x = 4, y = 4; ` ` ` `printSet(x, y); ` ` ` `return` `0; ` `} ` |

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Output:

0 4 1 3 2 2 3 1 4 0

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