Sudal's Garage

Question: Pentagonal 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. However, their difference, 70 − 22 = 48, is not pentagonal.Find the pair of pentagonal numbers, Pj and Pk, for which their sum and difference are pentagonal and D = |Pk − Pj| is minimised; what is..

Question: 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:d2d3d4=406 is divisible by 2d3d4d5=063 is divisible by 3d4d5d6=635 is divisible by 5d5d6d7=357 is divisibl..

Question: 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 value. For example, the word value for SKY is 19 + 11 + 25 = 55 = t10. If the word value is a triangle n..

Question: 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?문제: n 자리 팬디지털 수는 1부터 n까지 정확히 한 번씩만 써서 만든 수를 말한다. 예를 들어 2143은 4자리 팬디지털 수이면서 소수인 숫자이다.가장 큰 n 자리 팬디지털 소수를 구하여라. Solution: from itertools import permutations from sympy i..

Question: 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.d1 × d10 × d100 × d1000 × d10000 × d100000 × d1000000 문제: 양의 정수를 순서대로 이어붙여서 다음과 같은 무리수를 만들 수 있다.0.1234567891..