def S(p1, a): b = int(10 ** numLen(p1)) x, _ = extendedGCD(a, b) x *= p1 # y *= p1 t = x // b x -= t * b # y += t * a return a * x
def S(p1, a):
b = int(10**numLen(p1))
x, _ = extendedGCD(a, b)
x *= p1
# y *= p1
t = x // b
x -= t * b
# y += t * a
return a * x
def main(file):
# trick: for every anagram (word1, word2) calculate the character swaps
# required to turn word2 into word1 then perform those swaps on a square
# number of the same length as word1 and see if the resulting number is
# a square. Only caveat is that the eligible square numbers must have
# at most one of each digit
with open(file, 'r') as f:
words = f.read().replace('"', '').split(',')
# group the words into lists of anagrams
anagrams = dict()
for word in words:
anagrams.setdefault(''.join(sorted(word)), []).append(word)
# group the anagram lists by the length of the word
anagramsByLen = dict()
for k, v in anagrams.items():
if len(v) > 1:
anagramsByLen.setdefault(len(k), []).append(v)
# create eligible squares and group them by length
squaresByLen = dict()
for n in count(4):
sq = n * n
key = numLen(sq)
if key not in anagramsByLen:
break
if key == len(set(digits(sq))): # no repeated digits
squaresByLen.setdefault(key, []).append(sq)
def results():
for k in squaresByLen:
squares = squaresByLen[k]
for words in anagramsByLen[k]:
for word1, word2 in combinations(words, 2):
sw = list(swaps(word1, word2))
for sq in squares:
n = num(apply(digits(sq), sw))
if n in squares:
yield max(n, sq)
return max(results())
def main():
goodLast = lambda n: pandigital(n % 10**9)
goodFirst = lambda n: pandigital(n // 10**(numLen(n) - 9))
good = lambda n: n > 123456789 and goodLast(n) and goodFirst(n)
return next(k for k, n in enumerate(fibonaccis()) if good(n))
def dsum(n):
n = pow(Decimal(n), Decimal('0.5'))
n *= pow(10, 100 - numLen(int(n)))
n = int(n)
return sum(digits(n))
def main(limit): return next(n for n, f in enumerate(fibonaccis()) if numLen(f) >= limit)
def main(lim):
return sum(numLen(n) > numLen(d) for n, d in islice(convergents2(), lim))
def main(lim): return sum(numLen(n) > numLen(d) for n, d in islice(convergents2(), lim))
def main(): goodLast = lambda n: pandigital(n % 10 ** 9) goodFirst = lambda n: pandigital(n // 10 ** (numLen(n) - 9)) good = lambda n: n > 123456789 and goodLast(n) and goodFirst(n) return next(k for k, n in enumerate(fibonaccis()) if good(n))
def main(): # 10 ^ e has e + 1 digits, so max base is 9 # 9 ^ 22 has 21 digits, so 21 is the max exp return sum(numLen(pow(b, e)) == e for e in range(1, 22) for b in range(1, 10))
def main():
# 10 ^ e has e + 1 digits, so max base is 9
# 9 ^ 22 has 21 digits, so 21 is the max exp
return sum(
numLen(pow(b, e)) == e for e in range(1, 22) for b in range(1, 10))
def dsum(n):
n = pow(Decimal(n), Decimal('0.5'))
n *= pow(10, 100 - numLen(int(n)))
n = int(n)
return sum(digits(n))
def concat(a, b): return a * pow(10, numLen(b)) + b
def concat(a, b):
return a * pow(10, numLen(b)) + b
