Project Euler Solutions

What is Project Euler?

Note: All algorithms are run when "show solution" is first clicked, as a way of demonstrating computation time. Some functions are minified.

1. Multiples of 3 and 5

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

2. Even Fibonacci Numbers

Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ....

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.

3. Largest Prime Factor

The prime factors of 13195 are 5, 7, 13 and 29.

What is the largest prime factor of the number 600,851,475,143?

4. Largest Palindrome Product

A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 * 99.

Find the largest palindrome made from the product of two 3-digit numbers.

5. Smallest Multiple

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

6. Sum Square Difference

The sum of the squares of the first ten natural numbers is: 12 + 22 + ... + 102 = 385.

The square of the sum of the first ten natural numbers is:(1 + 2 + ... + 10)2 = 552 = 3025.

Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is: 3025 - 385 = 2640.

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

7. 10,001st Prime

By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6st prime is 13.

What is the 10,001st prime number?

8. Largest Product in a Series

The four adjacent digits in the following 1000-digit number that have the greatest product are 9 * 9 * 8 * 9 = 5832.

Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?

9. Special Pythagorean Triplet

A Pythagorean triplet is a set of three natural numbers, a < b < c, for which a2 + b2 = c2.

There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product a * b * c.

10. Summation of Primes

The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.

Find the sum of all the primes below two million.

11. Largest Product in a Grid

In the 20 by 20 grid below, four numbers along a diagonal line have been marked in red:

The product of these numbers is 26 * 63 * 78 * 14 = 1788696.

What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20 by 20 grid?

12. Highest Divisible Triangular Number

A sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

28 is the first triangle number to have over five divisors (divisible by 1, 2, 4, 7, 14, and 28).

What is the value of the first triangle number to have over five hundred divisors?

13. Large Sum

Below is one hundred 50-digit numbers:

What is the first ten digits of the sum of these numbers?

14. Longest Collatz Sequence

The following iterative sequence is defined for the set of positive integers:

n → n/2 (n is even)

n → 3n + 1 (n is odd)

Using the rule above and starting with 13, we generate the following sequence:

13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1

It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.

Which starting number, under one million, produces the longest chain? (NOTE: Once the chain starts the terms are allowed to go above one million.)

15. Lattice Paths

Starting in the top left corner of a 2 by 2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.

How many such routes are there through a 20 by 20 grid?

16. Power Digit Sum

215 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.

What is the sum of the digits of the number 21000?

17. Number Letter Counts

If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.

If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would be used? (not including spaces or hyphens)

18. Maximum Path Sum 1

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. Thus, given the following triangle:

What is the maximum total from top to bottom?

19. Counting Sundays

You are given the following information, but you may prefer to do some research for yourself:

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

20. Factorial Digit Sum

n! means n * (n - 1) * ... * 3 * 2 * 1. For example, 10! = 10 * 9 * ... * 3 * 2 * 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!

21. Amicable Numbers

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 != 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 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.

Evaluate the sum of all the amicable numbers under 10000.

22. Names Scores

Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand 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 * 53 = 49714.

What is the total of all the name scores in the file?

23. Non-abundant Sums

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.

A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

24. Lexicographic Permutations

A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are: [012, 021, 102, 120, 201, 210].

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

25. 1000-Digit Fibonacci Number

The Fibonacci sequence is defined by the recurrence relation: Fn = Fn-1 + Fn-2, where F1 = 1 and F2 = 1.

The 12th term, F12, is the first term to contain three digits (144).

What is the first term in the Fibonacci sequence to contain 1000 digits?