原题链接 http://projecteuler.net/problem=14

Longest Collatz sequence

The following iterative sequence is defined for the set of positive integers:

n ->n/2 (n is even)
n ->3n + 1 (n is odd)

Using the rule above and starting with 13, we generate the following sequence:
13 -> 40 ->20 ->10 ->5 ->16 -> 8 -> 4 ->2 ->1
It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.

Which starting number, under one million, produces the longest chain?

NOTE: Once the chain starts the terms are allowed to go above one million.

最长的考拉兹数:

在正整数上定义如下迭代序列:

n -> n / 2 (n是偶数)

n -> 3n + 1 (n是奇数)
从13开始,使用上面的规则,我们将得到如下序列:
13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
我们可以看到这个序列(从13开始到1结束)包含10个数。虽然这个还没有被证明(考拉兹问题),但我们可以认为所有的数都将在1结束。
求1 000 000以下的数,从哪一个数开始,产生的序列最长。
注意:一旦这个序列开始后,其中的数允许超过1 000 000。

解答:
如果将1到1000000都按照上述过程迭代,速度将会很慢,所以要保存一些计算结果,这样速度就会快很多了。
代码面前,了无秘密。直接上代码:

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#!/usr/bin/python
# -*- coding:utf-8 -*-
'''
Created on 2013-5-24

@author: shilong
@email: long470884130@163.com
'''

col = {}
col[1] = 1
def collatz(n):
if n in col:
return col[n]
else:
if n % 2 == 0:
col[n] = collatz(n / 2) + 1
else:
col[n] = collatz(3 * n + 1) + 1
return col[n]

原题连接 http://projecteuler.net/problem=13

Large sum

Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.
37107287533902102798797998220837590246510135740250
46376937677490009712648124896970078050417018260538
74324986199524741059474233309513058123726617309629
91942213363574161572522430563301811072406154908250
23067588207539346171171980310421047513778063246676
89261670696623633820136378418383684178734361726757
28112879812849979408065481931592621691275889832738
44274228917432520321923589422876796487670272189318
47451445736001306439091167216856844588711603153276
70386486105843025439939619828917593665686757934951
62176457141856560629502157223196586755079324193331
64906352462741904929101432445813822663347944758178
92575867718337217661963751590579239728245598838407
58203565325359399008402633568948830189458628227828
80181199384826282014278194139940567587151170094390
35398664372827112653829987240784473053190104293586
86515506006295864861532075273371959191420517255829
71693888707715466499115593487603532921714970056938
54370070576826684624621495650076471787294438377604
53282654108756828443191190634694037855217779295145
36123272525000296071075082563815656710885258350721
45876576172410976447339110607218265236877223636045
17423706905851860660448207621209813287860733969412
81142660418086830619328460811191061556940512689692
51934325451728388641918047049293215058642563049483
62467221648435076201727918039944693004732956340691
15732444386908125794514089057706229429197107928209
55037687525678773091862540744969844508330393682126
18336384825330154686196124348767681297534375946515
80386287592878490201521685554828717201219257766954
78182833757993103614740356856449095527097864797581
16726320100436897842553539920931837441497806860984
48403098129077791799088218795327364475675590848030
87086987551392711854517078544161852424320693150332
59959406895756536782107074926966537676326235447210
69793950679652694742597709739166693763042633987085
41052684708299085211399427365734116182760315001271
65378607361501080857009149939512557028198746004375
35829035317434717326932123578154982629742552737307
94953759765105305946966067683156574377167401875275
88902802571733229619176668713819931811048770190271
25267680276078003013678680992525463401061632866526
36270218540497705585629946580636237993140746255962
24074486908231174977792365466257246923322810917141
91430288197103288597806669760892938638285025333403
34413065578016127815921815005561868836468420090470
23053081172816430487623791969842487255036638784583
11487696932154902810424020138335124462181441773470
63783299490636259666498587618221225225512486764533
67720186971698544312419572409913959008952310058822
95548255300263520781532296796249481641953868218774
76085327132285723110424803456124867697064507995236
37774242535411291684276865538926205024910326572967
23701913275725675285653248258265463092207058596522
29798860272258331913126375147341994889534765745501
18495701454879288984856827726077713721403798879715
38298203783031473527721580348144513491373226651381
34829543829199918180278916522431027392251122869539
40957953066405232632538044100059654939159879593635
29746152185502371307642255121183693803580388584903
41698116222072977186158236678424689157993532961922
62467957194401269043877107275048102390895523597457
23189706772547915061505504953922979530901129967519
86188088225875314529584099251203829009407770775672
11306739708304724483816533873502340845647058077308
82959174767140363198008187129011875491310547126581
97623331044818386269515456334926366572897563400500
42846280183517070527831839425882145521227251250327
55121603546981200581762165212827652751691296897789
32238195734329339946437501907836945765883352399886
75506164965184775180738168837861091527357929701337
62177842752192623401942399639168044983993173312731
32924185707147349566916674687634660915035914677504
99518671430235219628894890102423325116913619626622
73267460800591547471830798392868535206946944540724
76841822524674417161514036427982273348055556214818
97142617910342598647204516893989422179826088076852
87783646182799346313767754307809363333018982642090
10848802521674670883215120185883543223812876952786
71329612474782464538636993009049310363619763878039
62184073572399794223406235393808339651327408011116
66627891981488087797941876876144230030984490851411
60661826293682836764744779239180335110989069790714
85786944089552990653640447425576083659976645795096
66024396409905389607120198219976047599490197230297
64913982680032973156037120041377903785566085089252
16730939319872750275468906903707539413042652315011
94809377245048795150954100921645863754710598436791
78639167021187492431995700641917969777599028300699
15368713711936614952811305876380278410754449733078
40789923115535562561142322423255033685442488917353
44889911501440648020369068063960672322193204149535
41503128880339536053299340368006977710650566631954
81234880673210146739058568557934581403627822703280
82616570773948327592232845941706525094512325230608
22918802058777319719839450180888072429661980811197
77158542502016545090413245809786882778948721859617
72107838435069186155435662884062257473692284509516
20849603980134001723930671666823555245252804609722
53503534226472524250874054075591789781264330331690

大整数求和

求这150个数相加得到的和的前10个数字.

解答:

这题没什么好说的,无非就是大整数求和,然后去前10个数字,如果用C++还有得写,用Python就很随意了。

原题链接 http://projecteuler.net/problem=12

Highly divisible triangular number

The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …

Let us list the factors of the first seven triangle numbers:

1: 1

3: 1,3

6: 1,2,3,6

10: 1,2,5,10

15: 1,3,5,15

21: 1,3,7,21

28: 1,2,4,7,14,28
We can see that 28 is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

三角数序列是由自然数相加形成的。第七个三角数是 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28.前十个三角数是:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …

我们列举出前7个三角形数的因子:

1: 1

3: 1,3

6: 1,2,3,6

10: 1,2,5,10

15: 1,3,5,10

21: 1,3,7,21

28: 1,2,4,7,14,28

我们可以看到28是第一个有超过5个因子的三角数

求第一个超过500个因子的三角数

解答:

这题的题意其实就是求素数因子,将三角数表示成素数因子乘积的形式,如 (28 = 2^2 * 7),这里素数因子2的次数的取值为0,1,2三种可能,素数因子7的次数的取值为0,1两种可能,所以28的因子有6个。按照这个思路,去做就可以了。

原题链接 http://projecteuler.net/problem=11

Largest product in a grid

In the 20 * 20 grid below, four numbers along a diagonal line have been marked in red.

08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91
22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21
24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48

The product of these numbers is 26 63 78 * 14 = 1788696.

What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20 * 20 grid?

在格子中最大的乘积

在下面的20 * 20的格子中,四个沿着对角线的数用红色标出

它们的乘积是 26 63 78 * 14 = 1788696.

求在这个20 * 20的格子中,四个相邻数的最大乘积(这四个相邻数必须在同一条直线上,向上,下,左,右,或者对角线)

解答:
这题没什么好说的,遍历。

原题链接 http://projecteuler.net/problem=10

Summation of primes

The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.

Find the sum of all the primes below two million.

素数的和

10以下的所有素数的和是 2 + 3 + 5 + 7 = 17.

求2000000以下所有素数的和。

解答:
用筛法求得2000000以下的所有素数,之后求和

写成代码如下:

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#!/usr/bin/python
# -*- coding:utf-8 -*-
'''
Created on 2013-5-5

@author: shilong
@email: long470884130@163.com
'''

from math import sqrt
def generate_prime(n):
'''得到[1,n]以内的素数'''
primes = [True for i in xrange(n + 1)]
primes[0] = primes[1] = False
for i in xrange(2,int(sqrt(n)) + 1):
if primes[i]:
s = i ** 2
while s <= n:
primes[s] = False
s += i
primes = [i for i in xrange(2,n + 1) if primes[i]]
return primes

if __name__ == "__main__":
n = 2000000
primes = generate_prime(n)
print sum(primes)

原题链接 http://projecteuler.net/problem=9

Special Pythagorean triplet

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
a2 + b2 = c2
For example, 32 + 42 = 9 + 16 = 25 = 52.

There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.

特殊的毕达哥拉斯三元组

一个毕达哥拉斯三元组指的是三个自然数, a < b < c ,且

a2 + b2 = c2

例如,32 + 42 = 9 + 16 = 25 = 52.

有且只有一组毕达哥拉斯三元组满足 a + b + c = 1000.

求abc的乘积

解答:

将 c = 1000 - a - b代入a2 + b2 = c2

得到1000 (a + b) = 500000 + a b

暴力找到满足这个条件的a和b.

原题链接 http://projecteuler.net/problem=8

Largest product in a series
Find the greatest product of five consecutive digits in the 1000-digit number.

73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450

连串数字的最大乘积
求这1000个数字中,连续5个数字的乘积的最大值

解答:
这题没什么好说的。

原题链接 http://projecteuler.net/problem=7

10001st prime

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.

What is the 10 001st prime number?

第10001个素数

列出前六个素数:2, 3,5, 7, 11,和13,我们可以知道第6个素数是13.

求第10001个素数。

解答:
没有想到什么好的方法,就用暴力解决。经过观察,对于大于6的正整数都可以用6n,6n + 1,6n + 2,6n + 3,6n + 4,6n + 5表示,其中只有6n + 1,6n + 5有可能是素数。所以可以用一个step变量来记录下一跳的步数,保证需要判断的数字在6n + 1和6n + 5中变换。要判断一个正整数是否是素数,只需用比它的平方根小的所有素数去除它,如果它可以被其中一个整除,则是素数,否则不是。

原题链接 http://projecteuler.net/problem=6

Sum square difference

The sum of the squares of the first ten natural numbers is,
12 + 22 + … + 102 = 385
The square of the sum of the first ten natural numbers is,
(1 + 2 + … + 10)2 = 552 = 3025
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

和平方差

前10个自然数的平方和是,

12 + 22 + … + 102 = 385

前10个自然数的和的平方是,

(1 + 2 + … + 10)2 = 552 = 3025

因此前10个自然数的和的平方与平方的和之间的差是 3025 - 385 = 2640.

求前100个自然数的和的平方与平方的和之间的差。

解答:

其实就是用公式

(1^2 + 2 ^2 + \ldots + n^2 = \frac{1}{6}n(n + 1)(2n + 1))

((1 + 2 + \ldots + n)^2 = (\frac{1}{2}n(n+1))^2)

原题链接http://projecteuler.net/problem=5

Smallest multiple

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

最小乘积

2520是能够被1到10整除的正整数中最小的。

求能够被1到20整除的正整数中最小的。

解答:

其实这题就是求1到10中的素数出现的最多次数,例如1到10中有素数2,3,5,7。其中2出现的次数最多为3次,即8;3出现的次数最多为2次,即9;其它为1次.所以最终的结果是8 9 5 * 7,即2520.

对于1到20,则有素数2,3,5,7,11,13,17,19.其中2出现的次数最多为4次,即16;3出现的次数为2,即9;其它都为 1次。所以最终的结果是16 9 5 7 11 13 17 * 19。