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 coinsint 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 methodint main(){ int N = 12; cout << findMaximumHeight(N) << endl; return 0;} |
Java
// Java program to find maximum height // of arranged coin triangleclass 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 coinsdef 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 methodN = 12print(findMaximumHeight(N))# This code is contributed by# Smitha Dinesh Semwal |
C#
// C# program to find maximum height // of arranged coin triangleusing 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. Notethat 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 coinsfunction 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 Codelet N = 12;document.write(findMaximumHeight(N)); // This code is contributed by avijitmondal1998.</script> |
Output:
4
Time complexity: O(log(n))
Auxiliary space: O(1)
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