# Euler problems/1 to 10

## Contents

## Problem 1

Add all the natural numbers below 1000 that are multiples of 3 or 5.

Solution:

```
problem_1 = sum [ x | x <- [1..999], (x `mod` 3 == 0) || (x `mod` 5 == 0)]
```

```
problem_1_v2 = sum $ filter (\x -> ( x `mod` 3 == 0 || x `mod` 5 == 0 ) ) [1..999]
```

```
sum1to n = n * (n+1) `div` 2
problem_1_v3 = sumStep 3 999 + sumStep 5 999 - sumStep 15 999
where sumStep s n = s * sum1to (n `div` s)
```

## Problem 2

Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million.

Solution:

```
problem_2 = sum [ x | x <- takeWhile (<= 1000000) fibs, x `mod` 2 == 0]
where fibs = 1 : 1 : zipWith (+) fibs (tail fibs)
```

The following two solutions use the fact that the even-valued terms in
the Fibonacci sequence themselves form a Fibonacci-like sequence
that satisfies
`evenFib 0 = 0, evenFib 1 = 2, evenFib (n+2) = evenFib n + 4 * evenFib (n+1)`

.

```
problem_2_v2 = sumEvenFibs $ numEvenFibsLessThan 1000000
sumEvenFibs n = (evenFib n + evenFib (n+1) - 2) `div` 4
evenFib n = round $ (2 + sqrt 5) ** (fromIntegral n) / sqrt 5
numEvenFibsLessThan n =
floor $ (log (fromIntegral n - 0.5) + 0.5*log 5) / log (2 + sqrt 5)
```

The first two solutions work because 10^6 is small. The following solution also works for much larger numbers (up to at least 10^1000000 on my computer):

```
problem_2_v3 = sumEvenFibsLessThan 1000000
sumEvenFibsLessThan n = (a + b - 1) `div` 2
where
n2 = n `div` 2
(a, b) = foldr f (0,1) $ takeWhile ((<= n2) . fst) $ iterate times2E (1, 4)
f x y | fst z <= n2 = z
| otherwise = y
where z = x `addE` y
addE (a, b) (c, d) = let ac = a*c in (a*d + b*c - 4*ac, ac + b*d)
times2E (a, b) = addE (a, b) (a, b)
```

## Problem 3

Find the largest prime factor of 317584931803.

Solution:

```
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
primeFactors n = factor n primes
where factor n (p:ps) | p*p > n = [n]
| n `mod` p == 0 = p : factor (n `div` p) (p:ps)
| otherwise = factor n ps
problem_3 = last (primeFactors 317584931803)
```

This can be improved by using
`null . tail`

instead of
`(== 1) . length`

.

## Problem 4

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

Solution:

```
problem_4 = foldr max 0 [ x | y <- [100..999], z <- [100..999], let x = y * z, let s = show x, s == reverse s]
```

An alternative to avoid evaluating twice the same pair of numbers:

```
problem_4' = foldr1 max [ x | y <- [100..999], z <- [y..999], let x = y * z, let s = show x, s == reverse s]
```

## Problem 5

What is the smallest number divisible by each of the numbers 1 to 20?

Solution:

```
problem_5 = head [ x | x <- [2520,5040..], all (\y -> x `mod` y == 0) [1..20]]
```

An alternative solution that takes advantage of the Prelude to avoid use of the generate and test idiom:

```
problem_5' = foldr1 lcm [1..20]
```

## Problem 6

What is the difference between the sum of the squares and the square of the sums?

Solution:

```
problem_6 = sum [ x^2 | x <- [1..100]] - (sum [1..100])^2
```

## Problem 7

Find the 10001st prime.

Solution:

```
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
primeFactors n = factor n primes
where factor n (p:ps) | p*p > n = [n]
| n `mod` p == 0 = p : factor (n `div` p) (p:ps)
| otherwise = factor n ps
problem_7 = head $ drop 10000 primes
```

As above, this can be improved by using
`null . tail`

instead of
`(== 1) . length`

.

Here is an alternative that uses a sieve of Eratosthenes:

```
primes' = 2 : 3 : sieve (tail primes') [5,7..]
where
sieve (p:ps) x = let (h, _:t) = span (p*p <) x
in h ++ sieve ps (filter (\q -> q `mod` p /= 0) t
problem_7_v2 = primes' !! 10000
```

## Problem 8

Discover the largest product of five consecutive digits in the 1000-digit number.

Solution:

```
num = ... -- 1000 digit number as a string
digits = map digitToInt num
groupsOf _ [] = []
groupsOf n xs = take n xs : groupsOf n ( tail xs )
problem_8 = maximum . map product . groupsOf 5 $ digits
```

## Problem 9

There is only one Pythagorean triplet, {*a*, *b*, *c*}, for which *a* + *b* + *c* = 1000. Find the product *abc*.

Solution:

```
problem_9 = head [a*b*c | a <- [1..500], b <- [a..500], let c = 1000-a-b, a^2 + b^2 == c^2]
```

Another solution using Pythagorean Triplets generation:

```
triplets :: Int -> [(Int, Int, Int)]
triplets l = [(a,b,c)|m <- [2..limit], n <- [1..(m-1)], let a = m^2 - n^2, let b = 2*m*n, let c = m^2 + n^2]
where limit = floor $ sqrt $ fromIntegral l
tripletWithLength :: Int -> [(Int, Int, Int)]
tripletWithLength n = filter ((==n) . f) $ triplets n
where
f (a,b,c) = a+b+c
problem_9 :: Int
problem_9 = prod3 $ head $ tripletWithLength 1000
where
prod3 (a,b,c) = a*b*c
```

## Problem 10

Calculate the sum of all the primes below one million.

Solution:

```
problem_10 = sum (takeWhile (< 1000000) primes)
```