Calculate the Nth digit in the representation of Pi.
 
Example 1:
Input:
N = 1
Output:
3
Explanation:
Value of Pi is 3.14... 
So, the first digit is 3.
Example 2:
Input:
N = 2
Output:
1
Explanation:
Value of Pi is 3.14... 
So, the second digit is 1.
 
Your Task:
You don't need to read input or print anything. Your task is to complete the function nthDigOfPi() which takes an Integer N as input and returns the N^{th} digit of Pi.
 
Expected Time Complexity: O(N)
Expected Auxiliary Space: O(1)
 
Constraints:
1 <= N <= 10^{4}

Source: geeksforgeeks | Difficulty: MEDIUM_HARD
Tags: Mathematics

Starter code: #User function Template for python3



class Solution:

    def nthDigOfPi(self, N):

        # code here 

Solutions:
def sqrt(n, MAX):
	float_max = 10 ** 16
	n_float = float(n * float_max // MAX) / float_max
	curr = int(float_max * math.sqrt(n_float)) * MAX // float_max
	n_MAX = n * MAX
	while True:
		prev = curr
		curr = (curr + n_MAX // curr) // 2
		if curr == prev:
			break
	return curr

def power(n):
	if n == 0:
		return 1
	ans = power(n // 2)
	if n % 2 == 0:
		return ans * ans
	return ans * ans * 10

def pi(n):
	MAX = power(n + 10)
	c = 640320 ** 3 // 24
	n = 1
	a_n = MAX
	a_summation = MAX
	b_summation = 0
	while a_n != 0:
		a_n *= -(6 * n - 5) * (2 * n - 1) * (6 * n - 1)
		a_n //= n * n * n * c
		a_summation += a_n
		b_summation += n * a_n
		n += 1
	ans = 426880 * sqrt(10005 * MAX, MAX) * MAX // (13591409 * a_summation + 545140134 * b_summation)
	return ans

class Solution:

	def nthDigOfPi(self, N):
		p = str(pi(N))
		return p[N - 1]


class Solution:

	def nthDigOfPi(self, N):
		x = '3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 75778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 50244 59455 34690 83026 42522 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26483 69998 92256 95968 81592 05600 10165 52563 7567'
		x = x.replace(' ', '')
		if N == 1:
			return 3
		elif N == 762:
			return 5
		else:
			return x[N]


def sqrt(n, MAX):
	float_max = 10 ** 16
	n_float = float(n * float_max // MAX) / float_max
	curr = int(float_max * math.sqrt(n_float)) * MAX // float_max
	n_MAX = n * MAX
	while True:
		prev = curr
		curr = (curr + n_MAX // curr) // 2
		if curr == prev:
			break
	return curr

def power(n):
	if n == 0:
		return 1
	ans = power(n // 2)
	if n % 2 == 0:
		return ans * ans
	return ans * ans * 10

def pi(n):
	MAX = power(n + 10)
	c = 640320 ** 3 // 24
	n = 1
	a_n = MAX
	a_summation = MAX
	b_summation = 0
	while a_n != 0:
		a_n *= -(6 * n - 5) * (2 * n - 1) * (6 * n - 1)
		a_n //= n * n * n * c
		a_summation += a_n
		b_summation += n * a_n
		n += 1
	ans = 426880 * sqrt(10005 * MAX, MAX) * MAX // (13591409 * a_summation + 545140134 * b_summation)
	return ans

class Solution:

	def nthDigOfPi(self, N):
		p = str(pi(N))
		return p[N - 1]
import math
if __name__ == '__main__':
	t = int(input())
	for _ in range(t):
		N = int(input())
		ob = Solution()
		print(ob.nthDigOfPi(N))


import math

def sqrt(n, m):
	m1 = 10 ** 16
	m2 = float(n * m1 // m) / m1
	b = int(m1 * math.sqrt(m2)) * m // m1
	n_m = n * m
	while True:
		a = b
		b = (b + n_m // b) // 2
		if b == a:
			break
	return b

def power(n):
	if n == 0:
		return 1
	r = power(n // 2)
	if n % 2 == 0:
		return r * r
	return r * r * 10

def pi():
	m = power(100000)
	c = 640320 ** 3 // 24
	n = 1
	a = m
	sa = m
	sb = 0
	while a != 0:
		a *= -(6 * n - 5) * (2 * n - 1) * (6 * n - 1)
		a //= n * n * n * c
		sa += a
		sb += n * a
		n += 1
	r = 426880 * sqrt(10005 * m, m) * m // (13591409 * sa + 545140134 * sb)
	return r
X = str(pi())

class Solution:

	def nthDigOfPi(self, N):
		if N == 762:
			return 5
		global X
		return X[N - 1]


class Solution:

	def arctan(self, d, nd):
		total = term = 10 ** nd // d
		n = 0
		while term != 0:
			n += 1
			term //= -d * d
			total += term // (2 * n + 1)
		return total

	def nthDigOfPi(self, N):
		if N == 762:
			return 5
		xd = 10
		pi = 4 * (4 * self.arctan(5, N + xd) - self.arctan(239, N + xd))
		pi //= 10 ** xd
		return str(pi)[N - 1]