def ans():
result = 0
p_max = 0
for D in range(2, 1000 + 1):
if is_square(D):
continue
limit = int(sqrt(D))
m = 0
d = 1
a = limit
num1 = 1
num = a
den1 = 0
den = 1
while num**2 - D * den**2 != 1:
m = d * a - m
d = (D - m * m) // d
a = (limit + m) // d
num2 = num1
num1 = num
den2 = den1
den1 = den
num = a * num1 + num2
den = a * den1 + den2
if num > p_max:
p_max = num
result = D
return result
def handle_one_side(w1: str, w2: str, digits: List[int]) -> int:
"""
use w1 to create a number for w2
:param w1: first word
:param w2: second word
:param digits: digits list
:return: a number found, -1 if not
"""
letters_values: Dict[str, int] = dict()
digs_used: Set[int] = set()
for index, c in enumerate(w1):
dig = digits[index]
# now set the value for the letter in c.
# if there is already a letter with this value, or c already has a value different than dig, break
if c in letters_values and letters_values[c] != dig:
return -1
if c not in letters_values and dig in digs_used:
return -1
# else, it is ok
letters_values[c] = dig
digs_used.add(dig)
else:
# if we did not break, calculate the value for the other word:
other_word_num_list = [letters_values[c] for c in w2]
# if it has a leading zero, go to next
if other_word_num_list[0] == 0:
return -1
other_word_num = list_to_num(other_word_num_list)
if is_square(other_word_num):
return other_word_num
return -1
def sum_n_digits(num: int, n: int) -> int:
"""
:param num: input number
:param n: number of digits
:return: 0 is perfect square, else sum of first n digits of sqrt(num)
"""
if is_square(num):
return 0
return digits_sum(square_root(num, n))
def ans():
"""
using calculus, the min distance for a,b,c if sqrt(a^2+(b-c)^2)
but we should also choose which of them is the a
because a>=b>=c, it is better to put a outside so the squares are more balanced
"""
count = 0
big_m = 0
while count < 10**6:
big_m += 1
a = big_m
for bc in range(2, 2 * a + 1):
x = bc**2 + a**2
if is_square(x):
count += bc // 2 - max(
bc - a - 1,
0) # add all pairs (b,c) such that b+c=bs, a>=b>=c>=1
return big_m
def square_root(num: int, digs: int) -> int:
"""
:param num: input number
:param digs: number of wanted digits
:return: first "digs" digits in the square root of num including digits before and after the decimal point
"""
if is_square(num):
return int(sqrt(num))
res = 0
for i in range(digs):
res *= 10
for j in range(1, 10):
# add until we pass the square
res += 1
np = num * 10**(2 * i)
if res**2 > np:
# too much, go back 1 and go for next digit to get more accurate
res -= 1
break
return res