Paper Cut into Minimum Number of Squares


Given a paper of size A x B. Task is to cut the paper into squares of any size. Find the minimum number of squares that can be cut from the paper.


Input  : 13 x 29
Output : 9
Explanation : 
2 (squares of size 13x13) + 
4 (squares of size 3x3) + 
3 (squares of size 1x1)=9

Input  : 4 x 5
Output : 5
Explanation : 
1 (squares of size 4x4) + 
4 (squares of size 1x1)

We know that if we want to cut minimum number of squares from the paper then we would have to cut largest square possible from the paper first and largest square will have same side as smaller side of the paper. For example if paper have the size 13 x 29, then maximum square will be of side 13. so we can cut 2 square of size 13 x 13 (29/13 = 2). Now remaining paper will have size 3 x 13. Similarly we can cut remaining paper by using 4 squares of size 3 x 3 and 3 squares of 1 x 1. So minimum 9 squares can be cut from the Paper of size 13 x 29.



Below is C++ implementation of above idea.

// C++ program to find minimum number of squares
// to cut a paper.
using namespace std;

// Returns min number of squares needed
int minimumSquare(int a, int b)
    long long result = 0, rem = 0;

    // swap if a is small size side .
    if (a < b)
        swap(a, b);

    // Iterate until small size side is
    // greater then 0
    while (b > 0)
        // Update result
        result += a/b;

        long long rem = a % b;
        a = b;
        b = rem;

    return result;

// Driver code
int main()
    int n = 13, m = 29;
    cout << minimumSquare(n, m);
    return 0;



Note that the above Greedy solution doesn’t always produce optimal result. For example if input is 36 x 30, the above algorithm would produce output 6, but we can cut the paper in 5 squares
1) Three squares of size 12 x 12
2) Two squares of size 18 x 18.

Thanks to Sergey V. Pereslavtsev for pointing out the above case.

This article is contributed by Kuldeep Singh(kulli_d_coder). If you like GeeksforGeeks and would like to contribute, you can also write an article using or mail your article to See your article appearing on the GeeksforGeeks main page and help other Geeks.

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