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Problem43

问题描述: 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 d1 be the 1st digit, d2 be the 2nd digit, and so on. In this way, we note the following: d2d3 ...

Problem 42

问题描述: The nth term of the sequence of triangle numbers is given by, tn = ½n(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 alphabetical position and adding these values we form a word valu ...

Problem 41

问题描述: 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? 解决问题:   这题和Problem24差不多。都是全排列的问题。 不过这里我们得找到所有的排列数字。然后判断是否是素数。 ...

Problem 40

算法描述: 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 dn represents the nth digit of the fractional part, find the value of the following expression. d1d1 ...

Problem 39

问题描述: 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 maximised?   解决问题:   public class Problem39 { public stati ...

Problem 38

问题描述: Take the number 192 and multiply it by each of 1, 2, and 3: 192 * 1 = 192192 * 2 = 384192 * 3 = 576 By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3) The same can be achieved by starting with 9 and mul ...

Problem 37

问题描述: 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, 379, 37, and 3. Find the sum of the only eleven primes that ...

Problem 36

问题描述: The 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 the palindromic number, in either base, may not include leading zeros.)   解决问题: java里Integer ...

Problem 35

问题描述: The 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, 37, 71, 73, 79, and 97. How many circular primes are there below one million?   解决问题:   先记录好 ...

Problem 34

问题描述: 145 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 they are not included.   解决问题:     public class Problem34 { public static final int UP ...

Problem 33

问题描述: The 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, is obtained by cancelling the 9s. We shall consider fractions like, 30/50 = 3/5, to be trivial examples. There are exactly fo ...

Problem 32

问题描述: 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, the 5-digit number, 15234, is 1 through 5 pandigital. The product 7254 is unusual, as the identity, 39 186 = 7254, containing multiplicand, multiplier, and product is 1 thr ...
问题描述: 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 + 150p + 220p + 15p + 12p + 31p How many different ways can £2 be made u ...

Problem 30

问题描述:   Surprisingly 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 + 44 As 1 = 14 is not a sum it is not included. The sum of these numbers is 1634 + 8208 + 9474 = 19316. Find ...

Problem 29

问题描述: Consider 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=3125 If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct t ...
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