Maximum height when coins are arranged in a triangle - GeeksforGeeks


Maximum height when coins are arranged in a triangle - GeeksforGeeks
We have N coins which need to arrange in form of a triangle, i.e. first row will have 1 coin, second row will have 2 coins and so on, we need to tell maximum height which we can achieve by using these N coins.

This problem can be solved by finding a relation between height of the triangle and number of coins. Let maximum height is H, then total sum of coin should be less than N,
Sum of coins for height H <= N
            H*(H + 1)/2  <= N
     H*H + H – 2*N <= 0
Now by Quadratic formula 
(ignoring negative root)

Maximum H can be (-1 + √(1 + 8N)) / 2 

Now we just need to find the square root of (1 + 8N) for
which we can use Babylonian method of finding square root
/* Returns the square root of n. Note that the function */
float squareRoot(float n)
{
    /* We are using n itself as initial approximation
      This can definitely be improved */
    float x = n;
    float y = 1;
 
    float e = 0.000001; /* e decides the accuracy level*/
    while (x - y > e)
    {
        x = (x + y) / 2;
        y = n/x;
    }
    return x;
}
 
//  Method to find maximum height of arrangement of coins
int findMaximumHeight(int N)
{
    //  calculating portion inside the square root
    int n = 1 + 8*N;
    int maxH = (-1 + squareRoot(n)) / 2;
    return maxH;
}
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