# Euler problems/11 to 20

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## Revision as of 13:53, 28 March 2007

## Contents |

## 1 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

## 2 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

## 3 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

## 4 Problem 14

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

Solution:

problem_14 = undefined

## 5 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

## 6 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 )

## 7 Problem 17

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

Solution:

problem_17 = undefined

## 8 Problem 18

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

Solution:

problem_18 = undefined

## 9 Problem 19

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

Solution:

problem_19 = undefined

## 10 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 ]