# Euler problems/41 to 50

### From HaskellWiki

(→[http://projecteuler.net/index.php?section=problems&id=48 Problem 48]: a solution) |
m (→[http://projecteuler.net/index.php?section=problems&id=48 Problem 48]) |
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− | If the problem were more computationally intensive, [http://en.wikipedia.org/wiki/Modular_exponentiation modular exponentiation] might be appropriate. As it is, ghci will return the result using the naive approach almost instantly. |
+ | If the problem were more computationally intensive, [http://en.wikipedia.org/wiki/Modular_exponentiation modular exponentiation] might be appropriate. With this problem size the naive approach is sufficient. |

<haskell> |
<haskell> |
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problem_48 = sum [n^n | n <- [1..1000]] `mod` 10^10 |
problem_48 = sum [n^n | n <- [1..1000]] `mod` 10^10 |

## Revision as of 01:57, 30 March 2007

## Contents |

## 1 Problem 41

What is the largest n-digit pandigital prime that exists?

Solution:

problem_41 = undefined

## 2 Problem 42

How many triangle words can you make using the list of common English words?

Solution:

problem_42 = undefined

## 3 Problem 43

Find the sum of all pandigital numbers with an unusual sub-string divisibility property.

Solution:

problem_43 = undefined

## 4 Problem 44

Find the smallest pair of pentagonal numbers whose sum and difference is pentagonal.

Solution:

problem_44 = undefined

## 5 Problem 45

After 40755, what is the next triangle number that is also pentagonal and hexagonal?

Solution:

problem_45 = head . dropWhile (<= 40755) $ match tries (match pents hexes) where match (x:xs) (y:ys) | x < y = match xs (y:ys) | y < x = match (x:xs) ys | otherwise = x : match xs ys tries = [n*(n+1) `div` 2 | n <- [1..]] pents = [n*(3*n-1) `div` 2 | n <- [1..]] hexes = [n*(2*n-1) | n <- [1..]]

## 6 Problem 46

What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?

Solution:

This solution is inspired by exercise 3.70 in *Structure and Interpretation of Computer Programs*, (2nd ed.).

problem_46 = head $ oddComposites `orderedDiff` gbSums oddComposites = filter ((>1) . length . primeFactors) [3,5..] gbSums = map gbWeight $ weightedPairs gbWeight primes [2*n*n | n <- [1..]] gbWeight (a,b) = a + b weightedPairs w (x:xs) (y:ys) = (x,y) : mergeWeighted w (map ((,)x) ys) (weightedPairs w xs (y:ys)) mergeWeighted w (x:xs) (y:ys) | w x <= w y = x : mergeWeighted w xs (y:ys) | otherwise = y : mergeWeighted w (x:xs) ys x `orderedDiff` [] = x [] `orderedDiff` y = [] (x:xs) `orderedDiff` (y:ys) | x < y = x : xs `orderedDiff` (y:ys) | x > y = (x:xs) `orderedDiff` ys | otherwise = xs `orderedDiff` ys

## 7 Problem 47

Find the first four consecutive integers to have four distinct primes factors.

Solution:

problem_47 = undefined

## 8 Problem 48

Find the last ten digits of 1^{1} + 2^{2} + ... + 1000^{1000}.

Solution: If the problem were more computationally intensive, modular exponentiation might be appropriate. With this problem size the naive approach is sufficient.

problem_48 = sum [n^n | n <- [1..1000]] `mod` 10^10

## 9 Problem 49

Find arithmetic sequences, made of prime terms, whose four digits are permutations of each other.

Solution:

problem_49 = undefined

## 10 Problem 50

Which prime, below one-million, can be written as the sum of the most consecutive primes?

Solution:

problem_50 = undefined