問題 The 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 are permutations of one another. There are n…

問題 The series, 1^1 + 2^2 + 3^3 + ... + 10^10 = 10405071317. Find the last ten digits of the series, 1^1 + 2^2 + 3^3 + ... + 1000^1000. ソース puts (1..1000).inject(0){|s, i| s + i**i} % (10 ** 10) 解答 9110846700 感想 ここまで来て１行プ…

問題 The first two consecutive numbers to have two distinct prime factors are: 14 = 2 × 7 15 = 3 × 5 The first three consecutive numbers to have three distinct prime factors are: 644 = 2^2 × 7 × 23 645 = 3 × 5 × 43 646 = 2 × 17 × 19. Find …

問題 It 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 + 2×1^2 15 = 7 + 2×2^2 21 = 3 + 2×3^2 25 = 7 + 2×3^2 27 = 19 + 2×2^2 33 = 31 + 2×1^2 It turns out tha…

問題 Triangle, pentagonal, and hexagonal numbers are generated by the following formulae: Triangle T(n)=n(n+1)/2 1, 3, 6, 10, 15, ... Pentagonal P(n)=n(3n−1)/2 1, 5, 12, 22, 35, ... Hexagonal H(n)=n(2n−1) 1, 6, 15, 28, 45, ... It can be ve…

問題 Pentagonal numbers are generated by the formula, P_(n)=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 P(4) + P(7) = 22 + 70 = 92 = P(8). However, their difference, 70 −…

問題 The 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. Let d(1) be the 1st digit, d(2) be the 2nd …

問題 The n'th term of the sequence of triangle numbers is given by, t(n) = 1/2 n(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 to a number corresponding to its alp…

問題 We 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-digit pandigital prime that exists? ソース cla…

問題 An 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 d(n) represents the n'th digit of the fraction…

問題 If 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 ≤ 1000, is the number of solutions maxim…

問題 Take the number 192 and multiply it by each of 1, 2, and 3: 192 × 1 = 192 192 × 2 = 384 192 × 3 = 576 By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1…

問題 The 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. Similarly we can work from right to left: 3797, 37…

問題 The decimal number, 585 = 1001001001_(2) (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 the palindromic number, in either base…