Project Euler

Problem Statement

You 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 thirty-one,
  Saving February alone,
  Which has twenty-eight, rain or shine.
  And on leap years, twenty-nine.
• A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400.

How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?

Problem Statement

\(n!\) means \(n \times (n - 1) \times … \times 3 \times 2 \times 1\).

For example, \(10! = 10 \times 9 \times … \times 3 \times 2 \times 1 = 3628800\), and the sum of the digits in the number \(10!\) is \(3 + 6 + 2 + 8 + 8 + 0 + 0 = 27\).

Find the sum of the digits in the number \(100!\)

Problem Statement

Let \(d(n)\) 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 \(a \neq b\), then \(a\) and \(b\) are an amicable pair and each of \(a\) and \(b\) are called amicable numbers.

For example, the proper divisors of \(220\) are \(1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110\); therefore \(d(220) = 284\). The proper divisors of \(284\) are \(1, 2, 4, 71, 142\); so \(d(284) = 220\).

#22 Names scores - Project Euler. projecteuler.net . Accessed Feb 18, 2022.

Problem Statement

Using names.txt, a 46K text file containing over 5,000 first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.

For example, when the list is sorted into alphabetical order, COLIN, which is worth \(3 + 15 + 12 + 9 + 14 = 53\), is the 938th name in the list. So COLIN would obtain a score of \(938 \times 53 = 49{,}714\).

#23 Non-abundant sums - Project Euler. projecteuler.net . Accessed Feb 19, 2023.

Problem Statement

A 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 \(1 + 2 + 4 + 7 + 14 = 28\), which means that \(28\) is a perfect number.