코딩 연습

A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before For example, 44 → 32 → 13 → 10 → 1 → 1 85 → 89 → 145 → 42 → 20 → 4 → 16 → 37 → 58 → 89 Therefore any chain that arrives at 1 or 89 will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at 1 or 89. How ma..

It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all. The square root of two is 1.41421356237309504880..., and the digital sum of the first one hundred decimal digits is 475. For the first one hundred natural numbers, find the total of the digital sums of ..

Let \(p(n)\) represent the number of different ways in which \(n\) coins can be separated into piles. For example, five coins can separated into piles in exactly seven different ways, so \(p(5)=7\). OOOOO OOOO O OOO OO OOO O O OO OO O OO O O O O O O O O Find the least value of \(n\) for which \(p(n)\) is divisible by one million. \(n\) 개의 동전을 각기 다른 방법으로 그룹 짓는 방법의 수를 \(p(n)\) 이라 하자. 예를 들어 5개의 동전에..

It is possible to write five as a sum in exactly six different ways: 4 + 1 3 + 2 3 + 1 + 1 2 + 2 + 1 2 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1 How many different ways can one hundred be written as a sum of at least two positive integers? 자연수 5를 두 자연수 이상의 합으로 나타내는 서로 다른 방법은 위와 같이 6가지가 있다. 자연수 100을 두 자연수 이상의 합으로 나타내는 서로 다른 방법은 몇 가지가 있을까? 이 문제를 풀기 위해서는 31번 문제를 잘 이해하고 와야 한다. 31번 문제를 이해했다면 76번은 31번과 유사한 문제로 변형..

31번 문제는 다음과 같다. In 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 the following way: 1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p How many different ways can £2 be made using any number of coins? 영국에는 다음과 같이 8가지 종류의 동전이 있다. 1펜스, 2펜스, 5펜스, 10펜스, 20펜스, 50펜스, 1..

72번 문제는 다음과 같다. Consider the fraction, n/d, where n and d are positive integers. If n

69번 문제는 다음과 같다. Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6. n Relatively Prime φ(n) n/φ(n) 2 1 1 2 3 1,2 2 1.5 4 1,3 2 2 5 1,2,3,4 4 1.25 6 1,5 2 3 7 1,2,3,4,5,6 6 1.1666... 8 1,3,5,7 4 2..

68번 문제는 다음과 같다. Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine. Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3. It is possible to complete the ..

67번 문제는 다음과 같다. By 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. 3 7 4 2 4 6 8 5 9 3 That is, 3 + 7 + 4 + 9 = 23. Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing a triangle with one-hundred rows. NOTE: It is not possible to try ..

66번 문제는 다음과 같다. Consider quadratic Diophantine equations of the form: x2 – Dy2 = 1 For example, when D=13, the minimal solution in x is 6492 – 13×1802 = 1. It can be assumed that there are no solutions in positive integers when D is square. By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the following: 32 – 2×22 = 1 22 – 3×12 = 1 92 – 5×42 = 1 52 – 6×22 = 1 82 – 7×32 = 1 Hen..