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原创 Project Euler Problem 57

Problem 57It is possible to show that the square root of two can be expressed as an infinite continued fraction.√ 2 = 1 + 1/(2 + 1/(2 + 1/(2 + … ))) = 1.414213…By expanding this for the first four ite...

2018-07-16 09:58:58 241

原创 Project Euler Problem 56

Problem 56A googol (10100) is a massive number: one followed by one-hundred zeros; 100100 is almost unimaginably large: one followed by two-hundred zeros. Despite their size, the sum of the digits in ...

2018-07-16 09:55:16 206

原创 Project Euler Problem 55

Problem 55If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.Not all numbers produce palindromes so quickly. For example,349 + 943 = 12921292 + 2921 = 42134213 + 3124 = 7337That is, 3...

2018-07-16 09:51:14 246

原创 Project Euler Problem 53

Problem 53There are exactly ten ways of selecting three from five, 12345:123, 124, 125, 134, 135, 145, 234, 235, 245, and 345In combinatorics, we use the notation, 5C3 = 10.In general,nCr=n!r!(n−r)!, ...

2018-07-16 09:48:15 249

原创 Project Euler Problem 52

Problem 52It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and...

2018-07-16 09:46:03 195

原创 Project Euler Problem 51

Problem 51By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.By replacing the 3rd and 4th digits of 56**3...

2018-07-16 09:42:14 237

原创 Project Euler Problem 50

Problem 50The prime 41, can be written as the sum of six consecutive primes:41 = 2 + 3 + 5 + 7 + 11 + 13This is the longest sum of consecutive primes that adds to a prime below one-hundred.The longest...

2018-07-16 09:37:53 140

原创 Project Euler Problem 49

Problem 49The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers ...

2018-07-16 09:35:00 186

原创 Project Euler Problem 48

Problem 48The series, 11 + 22 + 33 + … + 1010 = 10405071317.Find the last ten digits of the series, 11 + 22 + 33 + … + 10001000.十项的自幂级数求和为 11 + 22 + 33 + … + 1010 = 10405071317。求如下一千项的自幂级数求和的最后10位数字：1...

2018-06-27 23:37:48 332

原创 Project Euler Problem 47

Problem 47The first two consecutive numbers to have two distinct prime factors are:14 = 2 × 715 = 3 × 5The first three consecutive numbers to have three distinct prime factors are:644 = 22 × 7 × 23645...

2018-06-27 23:17:56 156

原创 Project Euler Problem 46

Problem 46It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.9 = 7 + 2×1215 = 7 + 2×2221 = 3 + 2×3225 = 7 + 2×3227 = 19 + 2×2...

2018-06-27 23:15:28 201

原创 Project Euler Problem 45

Problem 45Triangle, pentagonal, and hexagonal numbers are generated by the following formulae: TriangleTn=n(n+1)/21, 3, 6, 10, 15, …PentagonalPn=n(3n−1)/21, 5, 12, 22, 35, …HexagonalHn=n(2n−1)1, 6, ...

2018-06-27 23:14:11 208

原创 Project Euler Problem 44

Problem 44Pentagonal numbers are generated by the formula, Pn=n(3n−1)/2. The first ten pentagonal numbers are:1, 5, 12, 22, 35, 51, 70, 92, 117, 145, …It can be seen that P4 + P7 = 22 + 70 = 92 = P8. ...

2018-06-27 23:11:54 166

原创 Project Euler Problem 43

Problem 43The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property....

2018-06-21 23:10:50 183

原创 Project Euler Problem 42

Problem 42The nth term of the sequence of triangle numbers is given by, tn = 1/2n(n+1); so the first ten triangle numbers are:1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …By converting each letter in a word ...

2018-06-18 19:35:01 265

原创 Project Euler Problem 41

Problem 41We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.What is the largest n-dig...

2018-06-17 18:57:55 351

原创 Project Euler Problem40

Problem 40An irrational decimal fraction is created by concatenating the positive integers:0.123456789101112131415161718192021...It can be seen that the 12th digit of the fractional part is 1.If dn re...

2018-06-13 21:47:09 147

原创 Project Euler Problem 39

Problem 39If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.{20,48,52}, {24,45,51}, {30,40,50}For which value of p ≤ 10...

2018-06-08 20:12:56 206

原创 Project Euler Problem 38

Problem 38Take the number 192 and multiply it by each of 1, 2, and 3:192 × 1 = 192192 × 2 = 384192 × 3 = 576By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 19238457...

2018-06-02 20:35:29 169

原创 Project Problem 37

Problem 37The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. Simila...

2018-05-28 22:35:33 107

原创 Project Euler Problem 36

Problem 36The decimal number, 585 = 10010010012 (binary), is palindromic in both bases.Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2.(Please note that...

2018-05-25 20:53:53 133

原创 Project Euler Problem 35

Project 35The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31,...

2018-05-21 22:03:07 146

原创 Project Euler Problem 34

problem 34145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.Find the sum of all numbers which are equal to the sum of the factorial of their digits.Note: as 1! = 1 and 2! = 2 are not sums ...

2018-05-19 19:49:03 191

原创 Project Euler Problem 33

Problem 32Digit cancelling fractionsThe fraction 49/98 is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that 49/98 = 4/8, which is correct,...

2018-05-16 22:19:07 206

原创 Project Euler Problem 32

Problem 32We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once; for example, the 5-digit number, 15234, is 1 through 5 pandigital.The product 7254 is...

2018-05-15 21:21:45 206

原创 Project Euler Problem 31

Problem 31In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p).It is possible to make £2 in...

2018-05-12 19:57:49 186

原创 Project Euler Problem 30

Problem 30Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:1634 = 14 + 64 + 34 + 448208 = 84 + 24 + 04 + 849474 = 94 + 44 + 74 + 44As 1 = 14 is...

2018-05-10 21:13:25 172

原创 Project Euler Problem 29

Problem 29Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:22=4, 23=8, 24=16, 25=3232=9, 33=27, 34=81, 35=24342=16, 43=64, 44=256, 45=102452=25, 53=125, 54=625, 55=3125If they are t...

2018-05-09 21:06:15 169

原创 Project Euler Problem 28

Problem 28Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:21 22 23 24 2520 7 8 9 1019 6 1 2 1118 5 4 3 1217 16 15 14 13It can ...

2018-05-08 21:28:58 128

原创 Project Euler Problem 26

Problem 26A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:1/2= 0.51/3= 0.(3)1/4= 0.251/5= 0.21/6= 0.1(6)1/7= 0.(142857...

2018-05-07 21:52:33 301

原创 Project Euler Problem 27

Problem 27Euler discovered the remarkable quadratic formula:n2 + n + 41It turns out that the formula will produce 40 primes for the consecutive values n = 0 to 39. However, when n = 40, 402 + 40 + 41 ...

2018-05-06 21:40:30 287

原创 Project Euler Problem 25

Problem 25The Fibonacci sequence is defined by the recurrence relation:F1 = 1 F2 = 1Fn = Fn−1 + Fn−2Hence the first 12 terms will be:F1 = 1F2 = 1F3 = 2F4 = 3F5 = 5F6 = 8F7 = 13F8 = 21F9 = 34F10 = 55F...

2018-05-05 20:27:34 145

原创 Project Euler Problem 24

Problem 24A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetic...

2018-05-04 20:13:55 189

原创 Project Euler Problem 23

Problem 23A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, whic...

2018-05-03 21:51:09 170

原创 Project Euler Problem 22

Problem 22Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alpha...

2018-05-02 21:46:39 195

原创 Project Euler Problem 21

Problem 21Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a...

2018-05-01 21:11:03 109

原创 Project Euler Problem 20

Problem 20n! means n × (n − 1) × ... × 3 × 2 × 1For example, 10! = 10 × 9 × ... × 3 × 2 × 1 = 3628800,and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.Find the sum of the ...

2018-05-01 21:04:16 115

原创 Project Euler Problem 19

Problem 19You are given the following information, but you may prefer to do some research for yourself.1 Jan 1900 was a Monday.Thirty days has September,April, June and November.All the rest have thir...

2018-04-29 23:01:34 169

原创 Project Euler Problem 18

Problem 18By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.37 42 4 68 5 9 3That is, 3 + 7 + 4 + 9 = 23.Find the ...

2018-04-28 21:06:13 209 1

原创 Project Euler Problem 17

Problem 17If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.If all the numbers from 1 to 1000 (one thousand) incl...

2018-04-27 22:22:16 181

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