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.
Examples:
Input : N = 7 Output : 3 Maximum height will be 3, putting 1, 2 and then 3 coins. It is not possible to use 1 coin left. Input : N = 12 Output : 4 Maximum height will be 4, putting 1, 2, 3 and 4 coins, it is not possible to make height as 5, because that will require 15 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
Below code is implemented on above stated concept,
CPP
// C++ program to find maximum height of arranged // coin triangle #include <bits/stdc++.h> using namespace std;
/* 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;
} // Driver code to test above method int main()
{ int N = 12;
cout << findMaximumHeight(N) << endl;
return 0;
} |
Java
// Java program to find maximum height // of arranged coin triangle class GFG
{ /* Returns the square root of n.
Note that the function */
static float squareRoot( float n)
{
/* We are using n itself as
initial approximation.This
can definitely be improved */
float x = n;
float y = 1 ;
// e decides the accuracy level
float e = 0 .000001f;
while (x - y > e)
{
x = (x + y) / 2 ;
y = n / x;
}
return x;
}
// Method to find maximum height
// of arrangement of coins
static int findMaximumHeight( int N)
{
// calculating portion inside
// the square root
int n = 1 + 8 *N;
int maxH = ( int )(- 1 + squareRoot(n)) / 2 ;
return maxH;
}
// Driver code
public static void main (String[] args)
{
int N = 12 ;
System.out.print(findMaximumHeight(N));
}
} // This code is contributed by Anant Agarwal. |
Python3
# Python3 program to find # maximum height of arranged # coin triangle # Returns the square root of n. # Note that the function def squareRoot(n):
# We are using n itself as
# initial approximation
# This can definitely be improved
x = n
y = 1
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 def findMaximumHeight(N):
# calculating portion inside the square root
n = 1 + 8 * N
maxH = ( - 1 + squareRoot(n)) / 2
return int (maxH)
# Driver code to test above method N = 12
print (findMaximumHeight(N))
# This code is contributed by # Smitha Dinesh Semwal |
C#
// C# program to find maximum height // of arranged coin triangle using System;
class GFG
{ /* Returns the square root of n.
Note that the function */
static float squareRoot( float n)
{
/* We are using n itself as
initial approximation.This
can definitely be improved */
float x = n;
float y = 1;
// e decides the accuracy level
float e = 0.000001f;
while (x - y > e)
{
x = (x + y) / 2;
y = n / x;
}
return x;
}
static int findMaximumHeight( int N)
{
// calculating portion inside
// the square root
int n = 1 + 8*N;
int maxH = ( int )(-1 + squareRoot(n)) / 2;
return maxH;
}
/* program to test above function */
public static void Main()
{
int N = 12;
Console.Write(findMaximumHeight(N));
}
} // This code is contributed by _omg |
PHP
<?php // PHP program to find maximum height // of arranged coin triangle /* Returns the square root of n. Note that the function */ function squareRoot( $n )
{ /* We are using n itself as initial
approximation This can definitely
be improved */
$x = $n ;
$y = 1;
/* e decides the accuracy level*/
$e = 0.000001;
while ( $x - $y > $e )
{
$x = ( $x + $y ) / 2;
$y = $n / $x ;
}
return $x ;
} // Method to find maximum height of // arrangement of coins function findMaximumHeight( $N )
{ // calculating portion inside
// the square root
$n = 1 + 8 * $N ;
$maxH = (-1 + squareRoot( $n )) / 2;
return floor ( $maxH );
} // Driver code to test above method $N = 12;
echo findMaximumHeight( $N ) ;
// This code is contributed by anuj_67. ?> |
Javascript
<script> // Javascript program to find maximum height
// of arranged coin triangle /* Returns the square root of n.
Note that the function */
function squareRoot(n)
{
/* We are using n itself as
initial approximation.This
can definitely be improved */
let x = n;
let y = 1;
// e decides the accuracy level
let e = 0.000001;
while (x - y > e)
{
x = (x + y) / 2;
y = n / x;
}
return x;
}
// Method to find maximum height
// of arrangement of coins
function findMaximumHeight(N)
{
// calculating portion inside
// the square root
let n = 1 + 8*N;
let maxH = (-1 + squareRoot(n)) / 2;
return Math.round(maxH);
}
// Driver Code let N = 12; document.write(findMaximumHeight(N)); // This code is contributed by avijitmondal1998.
</script> |
Output:
4
Time complexity: O(log(n))
Auxiliary space: O(1)