# Euler problems/11 to 20

## Contents

## Problem 11

What is the greatest product of four numbers on the same straight line in the 20 by 20 grid?

Solution:

```
problem_11 = undefined
```

## Problem 12

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

Solution:

```
problem_12 = head $ filter ((> 500) . nDivisors) triangleNumbers
where triangleNumbers = scanl1 (+) [1..]
nDivisors n = product $ map ((+1) . length) (group (primeFactors n))
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 13

Find the first ten digits of the sum of one-hundred 50-digit numbers.

Solution:

```
nums = ... -- put the numbers in a list
problem_13 = take 10 . show . sum $ nums
```

## Problem 14

Find the longest sequence using a starting number under one million.

Solution:

```
problem_14 = undefined
```

## Problem 15

Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner?

Solution:

```
problem_15 = undefined
```

## Problem 16

What is the sum of the digits of the number 2^{1000}?

Solution:

```
dsum 0 = 0
dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d )
problem_16 = dsum ( 2^1000 )
```

## Problem 17

How many letters would be needed to write all the numbers in words from 1 to 1000?

Solution:

```
problem_17 = undefined
```

## Problem 18

Find the maximum sum travelling from the top of the triangle to the base.

Solution:

```
problem_18 = undefined
```

## Problem 19

How many Sundays fell on the first of the month during the twentieth century?

Solution:

```
problem_19 = undefined
```

## Problem 20

Find the sum of digits in 100!

Solution:

```
problem_20 = let fac n = product [1..n] in
foldr ((+) . Data.Char.digitToInt) 0 $ show $ fac 100
```

Alternate solution, summing digits directly, which is faster than the show, digitToInt route.

```
dsum 0 = 0
dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d )
problem_20' = dsum . product $ [ 1 .. 100 ]
```