直角等腰三角形可以容纳的最大正方形数

给定一个等腰(一个至少具有两个相等边的三角形)与底边b的直角三角形，我们需要找到边m的最大平方数，可以将其拟合到给定的三角形中。
例子：

Input : b = 6, m = 2
Output : 3

Input : b = 4, m = 1
Output : 6

让我们考虑一个直角三角形XYZ，其中YZ是三角形的底。假设底边的长度是b。如果考虑具有顶点Y的第一个正方形的位置，则在底面将有(b / m-1)个正方形，并且将剩下另一个底边长度为(b – m)的等腰直角三角形。
插图 ：

令f(b，m)=可以拟合为底长为b的三角形的平方数。
然后f(b，m)=(b / m – 1)+ f(b – m，m)
我们可以使用上述递归和记忆来计算f(b)。稍后，我们可以在O(1)时间内回答每个查询。如果(b <2 * m)f(b，m)= 0，我们可以用基本情况分别对偶数和奇数进行处理。
给定的递归可以解决为：
f(b，m)= b / m – 1 + f(b – m，m)= b / m – 1 +(b – m)/ m – 1 + f(b – 2m，m)
f(b，m)= b / m – 1 + b / m – 2 + f(b – 3m，m)+…+ f(b –(b / m)m，m)
f(b)= b / m – 1 + b / m – 2 + b / m – 3 +….. + 1 + 0
在有条件的情况下，如果(b <2 * m)f(b，m)= 0
f(b)=第一个(b / m – 1)自然数之和
=(b / m – 1)*(b / m)/ 2
该公式可用于将时间复杂度降低到O(1)。

C++

// CPP program for finding maximum squares
// that can fit in right angle isosceles
// triangle
#include
using namespace std;
 
// function for finding max squares
int maxSquare(int b, int m)
{
    // return in O(1) with derived
    // formula
    return (b / m - 1) * (b / m) / 2;
}
 
// driver program
int main()
{
    int b = 10, m = 2;
    cout << maxSquare (b,m);
    return 0;
}

Java

// Java program for finding maximum squares
// that can fit in right angle isosceles
// triangle
public class GFG
{    
    // function for finding max squares
    static int maxSquare(int b, int m)
    {
        // return in O(1) with derived
        // formula
        return (b / m - 1) * (b / m) / 2;
    }
      
    // driver program
    public static void main(String args[])
    {
        int b = 10, m = 2;
        System.out.println(maxSquare (b,m));
    }
}
 
// This code is contribute by Sumit Ghosh

Python3

# Python3 program for
# finding maximum squares
# that can fit in
# right angle isosceles
# triangle
 
# function for finding max squares
def maxSquare(b, m):
  
    # return in O(1) with derived
    # formula
    return (b / m - 1) * (b / m) / 2
  
 
# driver program
b = 10
m = 2
print(int(maxSquare (b,m)))
 
# This code is contributed by
# Smitha Dinesh Semwal

C#

// C# program for finding maximum squares
// that can fit in right angle isosceles
// triangle
using System;
 
public class GFG
{
    // function for finding max squares
    static int maxSquare(int b, int m)
    {
        // return in O(1) with derived
        // formula
        return (b / m - 1) * (b / m) / 2;
    }
     
    // driver program
    public static void Main()
    {
        int b = 10, m = 2;
        Console.WriteLine(maxSquare (b, m));
    }
}
 
// This code is contribute by vt_m

Javascript

 
// Javascript program for finding maximum squares
// that can fit in right angle isosceles
// triangle
 
// function for finding max squares
function maxSquare(b, m)
{
 
    // return in O(1) with derived
    // formula
    return (b / m - 1) * (b / m) / 2; a
}
 
// Driver program
  
    let b = 10, m = 2;
    document.write(maxSquare (b,m));
     
// This code is contributed by Mayank Tyagi

输出：

10