Python Program for Maximum height when coins are arranged in a triangle

Last Updated : 21 Feb, 2023

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
Explanation: 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
Explanation: 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.

Brute-Force Approach:
A triangle takes coins in increasing order i.e., first line will have 1 coin, second line will have 2 coins and so on.
So we can subtract [1.2.3.,,,] till n is greater than the number, and increment our answer.

Below is the implementation of the above approach:

Python3

def triangle(input1):     if input1 < 1:         return 'Zero'     else:           # Counter for controlling the coins in a row         a = 1                 height = 0         while input1 > 0:             if input1 - a >= 0:                 input1 = input1 - a                 a = a + 1                 height = height + 1             else:                 break         return height   if __name__ == "__main__":     testinput1 = 22     print(triangle(testinput1))

Output

Time complexity: O(sqrt(n))
Auxiliary space: O(1).

Efficient Approach: 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

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