def isOct(n):
if not isPerfectSquare(4+12*n):
return False
num = int(sqrt(4 + 12*n))
if (num + 4) % 4 == 0:
return True
return False
def isPent(n):
if not isPerfectSquare(1 + 24 * n):
return False
num = int(sqrt(1 + 24 * n))
if (num + 1) % 6 == 0:
return True
return False
def isHept(n):
if not isPerfectSquare(9+40*n):
return False
num = int(sqrt(9 + 40*n))
if (num + 3) % 10 == 0:
return True
return False
def isHex(n):
if not isPerfectSquare(1+8*n):
return False
num = int(sqrt(1 + 8*n))
if (num + 1) % 4 == 0:
return True
return False
def paths(M):
count = 0
for c in range(3, M + 1):
for ab in range(3, 2 * c):
if c % 2 == 1 and ab % 2 == 1:
continue
elif isPerfectSquare(ab**2 + c**2):
if ab < c:
count += ab // 2
else:
count += c + 1 - (ab + 1) // 2
return count
# Project Euler
# Problem 206
# Concealed square
import sys
sys.path.append('..')
from euler import isPerfectSquare
from itertools import product
l = [str(x) for x in range(10)]
num = '1234567890'
def splice(str1, str2):
result = str1[0]
for i in range(len(str2)):
result += str2[i] + str1[i + 1]
return result
for p in product(l, repeat=9):
string = splice(num, ''.join(p))
if isPerfectSquare(int(string)):
break
print(int(int(string)**(1 / 2)))
from euler import isPerfectSquare
from math import sqrt
import time
'''
Need to solve
b(b-1) / n(n-1) = 1/2
2b**2 - 2b = n**2 - n
2(b - 1/2)**2 = n**2 - n + 1/2
b = 0.5 + 0.5 * sqrt(n**2 - n + 1/2)
'''
start = time.time()
target = 10**1
n = target
x = int(target * sqrt(2))
x += 1 - (x % 2) # make x odd only
while not isPerfectSquare(2 * x**2 - 1):
x += 2
#print(x)
n = int(0.5 * (1 + sqrt(2 * x**2 - 1)) + 1 / 2) # + 1/2 for rounding
print('n = ' + str(n))
b = int(0.5 * (1 + sqrt(2 * n**2 - 2 * n + 1)) + 1 / 2)
print('b = ' + str(b))
print(b * (b - 1) / (n * (n - 1)))
print('time taken: ' + str(time.time() - start) + ' seconds')
# Problem 80
# Square root digital expansion
import sys
sys.path.append('..')
from euler import isPerfectSquare
'''
Calculates the sum of the first d digits of sqrt(num)
'''
def sqrtDecimalSum(num, d):
target = num
n = 1
for i in range(d):
while (n + 1)**2 < target:
n += 1
n *= 10
target *= 100
return str(n)
ans = 0
for n in range(1, 101):
if not isPerfectSquare(n):
ans += sum([int(x) for x in sqrtDecimalSum(n, 100)])
print(ans)
def isSquare(n):
if not isPerfectSquare(n**2):
return False
return True
