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

This article is contributed by **Utkarsh Trivedi**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the **DSA Self Paced Course** at a student-friendly price and become industry ready. To complete your preparation from learning a language to DS Algo and many more, please refer **Complete Interview Preparation Course****.**

In case you wish to attend live classes with industry experts, please refer **DSA Live Classes**