Project Euler 50-56

Project Euler: https://projecteuler.net/
Project Euler | 欧拉计划: https://pe-cn.github.io/
Project Euler 1-7
Project Euler 8-14
Project Euler 15-21 & 67
Project Euler 22-28
Project Euler 29-35
Project Euler 36-42
Project Euler 43-49
Project Euler 50-56
Project Euler 57-63
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Problem 50: Consecutive prime sum

Which prime, below one-million, can be written as the sum of the most consecutive primes?

def eratosthenes_sieve(max_number):sieve = [0] * (max_number + 1)sieve[0], sieve[1] = -1, -1for i in range(2, max_number + 1):if sieve[i] == 0:for j in range(2 * i, max_number + 1, i):sieve[j] = 1return sieve
primes = []
for i, elem in enumerate(eratosthenes_sieve(1000001)):if elem == 0: # 从题目示例开始查找primes.append(i)length = 1
s = primes[0]
while True:s += primes[length]if s >= 1000000: breaklength += 1        while length > 1:Sum = sum(primes[:length])if Sum in primes: breaki = 0found = Falsewhile not found:Sum += primes[length + i] - primes[i]if Sum >= 1000000: breakfound = Sum in primesi += 1if found: breaklength -= 1
print(Sum)

Problem 51: Prime digit replacements

Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family.

counter: 一个计数器工具提供快速和方便的计数。
https://docs.python.org/zh-cn/3/library/collections.html#collections.Counter
-(1) 至少要是一个四位数，我们可以从数字1111开始搜寻；
-(2) 重复的数位只能是三或三的倍数；
-(3) 重复的数字只能是0, 1, 2三个数。
参考https://www.cnblogs.com/metaquant/p/11820821.html

def eratosthenes_sieve(max_number):sieve = [0] * (max_number + 1)sieve[0], sieve[1] = -1, -1for i in range(2, max_number + 1):if sieve[i] == 0:for j in range(2 * i, max_number + 1, i):sieve[j] = 1return sieve
primes = []
for i, elem in enumerate(eratosthenes_sieve(1000001)):if elem == 0 and i>1111: primes.append(i)

from collections import Counterdef is_replacable_prime(n):s = str(n)count = Counter(s)num,d = count.most_common(1)[0]if d % 3 == 0 and num in set('012'):k = 1for j in range(int(num)+1,10):new = s.replace(num,str(j))if int(new) in primes:k += 1if k == 8:return Truereturn Falsedef main():for p in primes:if is_replacable_prime(p):return p
main()

Problem 52: Permuted multiples

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits.

for a in range(1,1000000):b = sorted(str(a))for i in range(2, 7):if len(str(a*i)) > len(b):breakif b != sorted(str(a*i)):breakelse:print(a)break

Problem 53: Combinatoric selections

How many, not necessarily distinct, values of nCr, for 1 ≤ n ≤ 100, are greater than one-million?

from itertools import combinations
import math
def C(n,m): # 计算组合数return math.factorial(n)//(math.factorial(m)*math.factorial(n-m))ans = 0
for i in range(23,101):for j in range(1,i):if C(i,j)>1000000:ans += 1
print(ans)

Problem 54：Poker hands

How many hands does Player 1 win?

参考j123 https://projecteuler.net/best_posts=054

T=(([1],[3,1.5]),([3,1.7],[5]))# 烂牌，顺子，同花，同花顺
def rank(H, v=dict(zip(b'23456789TJQKA', range(13)))):C,V,S = zip(*reversed(sorted([(H.count(c), v[c], s) for c, s in H.split()])))return ([C[0], C[C[0]]] if C[0]>1 else T[len(set(S))==1][V[0]-V[4]==4]), Vprint(sum([rank(s[:14]) > rank(s[15:]) for s in open('data/p054_poker.txt', 'rb')]))

Problem 55：Lychrel numbers

How many Lychrel numbers are there below ten-thousand?

def is_palindrome(n):n = str(n)if n == n[::-1]:return Truereturn Falsedef is_lychrel(n):for i in range(50):n += int(str(n)[::-1])if is_palindrome(n):return Falsereturn Trueans = 0
for i in range(1, 10001):if is_lychrel(i):ans += 1
print(ans)

Problem 56: Powerful Digit Sum

Considering natural numbers of the form, a b a^b ab, where a, b < 100, what is the maximum digital sum?

大数乘法

ans = []
for a in range(2,100):NUM = [1]carry = 0for b in range(2,100):carry = 0for j in range(len(NUM)):temp = NUM[j] * a + carryif temp>9:NUM[j] = temp%10carry = temp//10else:carry = 0NUM[j] = temp      while carry>0:temp = carryNUM.append(temp%10)carry = temp//10ans.append(sum(NUM))
print(max(ans))

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欧拉计划 Project Euler 50-56