Hướng dẫn coin triangle python code

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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.

def squareRoot(n):

x = n

y = 1

e = 0.000001

while (x - y > e):

x = (x + y) / 2

y = n/x

return x

def findMaximumHeight(N):

n = 1 + 8*N

maxH = (-1 + squareRoot(n)) / 2

return int(maxH)

N = 12

print(findMaximumHeight(N))

Output:

Please refer complete article on Maximum height when coins are arranged in a triangle for more details!

Pascal’s triangle is a pattern of the triangle which is based on nCr, below is the pictorial representation of Pascal’s triangle.

Example:

Input: N = 5
Output:
      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1

Method 1: Using nCr formula i.e. n!/(n-r)!r!

After using nCr formula, the pictorial representation becomes:

          0C0
       1C0   1C1
    2C0   2C1   2C2
 3C0   3C1   3C2    3C3

Algorithm:

Take a number of rows to be printed, lets assume it to be n
Make outer iteration i from 0 to n times to print the rows.
Make inner iteration for j from 0 to (N – 1).
Print single blank space ” “.
Close inner loop (j loop) //its needed for left spacing.
Make inner iteration for j from 0 to i.
Print nCr of i and j.
Close inner loop.
Print newline character (\n) after each inner iteration.

Implementation:

Python3

from math import factorial

n = 5

for i in range(n):

for j in range(n-i+1):

print(end=" ")

for j in range(i+1):

print(factorial(i)//(factorial(j)*factorial(i-j)), end=" ")

print()

Output:

Time complexity: O(N2)

Method 2: We can optimize the above code by the following concept of a Binomial Coefficient, the i’th entry in a line number line is Binomial Coefficient C(line, i) and all lines start with value 1. The idea is to calculate C(line, i) using C(line, i-1).

C(line, i) = C(line, i-1) * (line - i + 1) / i

Implementations:

Python3

n = 5

for i in range(1, n+1):

for j in range(0, n-i+1):

print(' ', end='')

C = 1

for j in range(1, i+1):

print(' ', C, sep='', end='')

C = C * (i - j) // j

print()

Output:

Time complexity: O(N2)

Auxiliary Space: O(1) because it is using constant space for variables

Method 3: This is the most optimized approach to print Pascal’s triangle, this approach is based on powers of 11.

11**0 = 1
11**1 = 11
11**2 = 121
11**3 = 1331

Implementation:

Python3

n = 5

for i in range(n):

print(' '*(n-i), end='')

print(' '.join(map(str, str(11**i))))

Output:

Time Complexity: O(N)

Auxiliary Space: O(1) as using constant variable

However, this approach is applicable up to n=5 only.

Hướng dẫn coin triangle python code

Python3

Python3

Python3

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