def all_base10_base2_palindromes(n):
result = []
base_10 = 1
base_2 = '1'
while base_10 < n:
if is_palindrome(base_10) and is_palindrome(base_2):
result.append(base_10)
base_10 += 1
base_2 = binary_incrementer(base_2)
return result
def palindromic_square_sums(n):
# first populate all pairs that add to less than n
# 2k**2 < k**2 + (k + 1)**2 < n
MAX_k = int(round(sqrt(n / 2.0)))
curr = [index ** 2 + (index + 1) ** 2 for index in range(1, MAX_k)]
curr = [num for num in curr if num < n]
result = [num for num in curr if is_palindrome(num)]
num_squares = 2
while curr:
num_squares += 1
curr = [curr[i] + (i + num_squares) ** 2 for i in range(len(curr))]
curr = [num for num in curr if num < n]
result.extend([num for num in curr if is_palindrome(num)])
return set(result)
def main(verbose=False):
lychrel_sequences = {}
for i in range(1, 10000):
update_hash(i, 50, lychrel_sequences)
# We begin with the inital assumption that every number
# is a Lychrel number, and reduce the count every time
# we encounter a number which is not
count = 9999
for key, value in lychrel_sequences.items():
if 50 in value:
iterations = 50
else:
iterations = max(value)
if is_palindrome(value[iterations]):
count -= 1
return count
def update_hash(n, max_iterations, hash_={}):
"""
Uses the hash values and continually updates
the sequence until a palindrome is found or until
the number of iterations exceeds max_iterations
"""
curr = next_lychrel_value(n)
to_add = {0: n, 1: curr}
index = 1
while not is_palindrome(curr) and index <= max_iterations:
if curr in hash_:
covered = hash_[curr].copy()
for i in range(1, max(covered) + 1):
to_add[index + i] = covered[i]
index += max(covered)
else:
curr = next_lychrel_value(curr)
index += 1
to_add[index] = curr
hash_[n] = to_add
return to_add
def main(verbose=False):
products = apply_to_list(operator.mul, range(100, 1000))
return max(elt for elt in products if is_palindrome(elt))