The task is simply stated. Given an integer n (3 < n < 10^(9)), find the length of the smallest list of [*perfect squares*](https://en.wikipedia.org/wiki/Square_number) which add up to n. Come up with the best algorithm you can; you'll need it!

Examples:

sum_of_squares(17) = 2  17 = 16 + 1 (4 and 1 are perfect squares).
sum_of_squares(15) = 4  15 = 9 + 4 + 1 + 1. There is no way to represent 15 as the sum of three perfect squares.
sum_of_squares(16) = 1  16 itself is a perfect square.

Time constraints:

5 easy (sample) test cases: n < 20

5 harder test cases: 1000 < n < 15000

5 maximally hard test cases: 5 * 1e8 < n < 1e9

```if:java
300 random maximally hard test cases: 1e8 < n < 1e9
```
```if:c#
350 random maximally hard test cases: 1e8 < n < 1e9
```
```if:python
15 random maximally hard test cases: 1e8 < n < 1e9
```
```if:ruby
25  random maximally hard test cases: 1e8 < n < 1e9
```
```if:javascript
100 random maximally hard test cases: 1e8 < n < 1e9
```
```if:crystal
250 random maximally hard test cases: 1e8 < n < 1e9
```
```if:cpp
Random maximally hard test cases: 1e8 < n < 1e9
```

Source: codewars | Difficulty: MEDIUM
Tags: Mathematics

Starter code: def sum_of_squares(n):
	

Solutions:
def one_square(n):
	return round(n ** 0.5) ** 2 == n

def two_squares(n):
	while n % 2 == 0:
		n //= 2
	p = 3
	while p * p <= n:
		while n % (p * p) == 0:
			n //= p * p
		while n % p == 0:
			if p % 4 == 3:
				return False
			n //= p
		p += 2
	return n % 4 == 1

def three_squares(n):
	while n % 4 == 0:
		n //= 4
	return n % 8 != 7

def sum_of_squares(n):
	if one_square(n):
		return 1
	if two_squares(n):
		return 2
	if three_squares(n):
		return 3
	return 4


sq = lambda n: n ** 0.5 % 1 == 0

def sum_of_squares(n):
	if sq(n):
		return 1
	while not n & 3:
		n >>= 2
	if n & 7 == 7:
		return 4
	for i in range(int(n ** 0.5) + 1):
		if sq(n - i * i):
			return 2
	return 3


from math import sqrt

def sum_of_squares(n):
	n_sqrt = int(sqrt(n))
	if n == n_sqrt * n_sqrt:
		return 1
	(div, mod) = divmod(n, 4)
	while mod == 0:
		(div, mod) = divmod(div, 4)
	if (div * 4 + mod) % 8 == 7:
		return 4
	for i in range(1, n_sqrt + 1):
		temp = n - i * i
		temp_sqrt = int(sqrt(temp))
		if temp == temp_sqrt * temp_sqrt:
			return 2
	return 3


from collections import Counter
sum_of_squares = lambda n: 1 if (n ** 0.5).is_integer() else 2 if all((not j & 1 for (i, j) in Counter(two(n)).items() if i % 4 == 3)) else 4 if four(n) else 3

def two(n):
	(j, li) = (2, [])
	while j * j <= n:
		if n % j:
			j += 1
			continue
		li.append(j)
		n //= j
	return li + [[], [n]][n > 0]

def four(x):
	while x % 4 == 0:
		x //= 4
	return 8 * (x // 8) + 7 == x


import math

def sum_of_squares(n):
	final = []
	de = 0
	if str(math.sqrt(n))[::-1].find('.') == 1:
		return 1
	else:
		for i in range(int(n ** 0.5)):
			ans = []
			de = n - int(de ** 0.5 - i) ** 2
			ans.append(int(de ** 0.5))
			while de > 0:
				de -= int(de ** 0.5) ** 2
				ans.append(int(de ** 0.5))
			final.append(len(ans))
		return min(final)


def sum_of_squares(n):
	import math
	while n % 4 == 0:
		n = n // 4
	if n % 8 == 7:
		return 4
	if int(math.sqrt(n)) ** 2 == n:
		return 1
	i = 1
	while i * i <= n:
		j = math.sqrt(n - i * i)
		if int(j) == j:
			return 2
		i += 1
	return 3


def sum_of_squares(n):

	def is_sq(n):
		return int(n ** 0.5) * int(n ** 0.5) == n
	if is_sq(n):
		return 1
	while n & 3 == 0:
		n >>= 2
	if n & 7 == 7:
		return 4
	for i in range(1, int(n ** 0.5)):
		if is_sq(n - i * i):
			return 2
	return 3


def sum_of_squares(n):

	def divide_count(p):
		nonlocal n, s1, s2
		while not n % p ** 2:
			n //= p * p
		if not n % p:
			s1 &= False
			s2 &= p & 3 < 3
			n //= p
	s1 = s2 = True
	divide_count(2)
	s3 = n << 1 - s1 & 7 < 7
	divide_count(3)
	(x, wheel) = (5, 2)
	while x * x <= n:
		divide_count(x)
		(x, wheel) = (x + wheel, 6 - wheel)
	if n > 1:
		divide_count(n)
	return 4 - s3 - s2 - s1


def sum_of_squares(n):
	if n ** 0.5 == int(n ** 0.5):
		return 1
	while n % 4 == 0:
		n >>= 2
	if (n - 7) % 8 == 0:
		return 4
	for s in range(1, int(n ** 0.5) + 1):
		if (n - s * s) ** 0.5 == int((n - s * s) ** 0.5):
			return 2
	return 3


import math

def is_perfect_sq(n):
	sr = math.sqrt(n)
	low_perfect = math.floor(sr)
	check = sr - low_perfect == 0
	return check

def sum_of_squares(n):
	if is_perfect_sq(n):
		return 1
	while n % 4 == 0:
		n >>= 2
	if n % 8 == 7:
		return 4
	for a in range(math.ceil(math.sqrt(n))):
		b = n - a * a
		if is_perfect_sq(b):
			return 2
	return 3