# Euler problems/41 to 50

## Contents

## Problem 41

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

Solution:

```
problem_41 = head [p | n <- init (tails "987654321"),
p <- perms n, isPrime (read p)]
where perms [] = [[]]
perms xs = [x:ps | x <- xs, ps <- perms (delete x xs)]
isPrime n = n > 1 && smallestDivisor n == n
smallestDivisor n = findDivisor n (2:[3,5..])
findDivisor n (testDivisor:rest)
| n `mod` testDivisor == 0 = testDivisor
| testDivisor*testDivisor >= n = n
| otherwise = findDivisor n rest
```

## Problem 42

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

Solution:

```
score :: String -> Int
score = sum . map ((subtract 64) . ord . toUpper)
istrig :: Int -> Bool
istrig n = istrig' n trigs
istrig' :: Int -> [Int] -> Bool
istrig' n (t:ts) | n == t = True
| otherwise = if t < n && head ts > n then False else istrig' n ts
trigs = map (\n -> n*(n+1) `div` 2) [1..]
--get ws from the Euler site
ws = ["A","ABILITY" ... "YOURSELF","YOUTH"]
problem_42 = length $ filter id $ map (istrig . score) ws
```

## Problem 43

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

Solution:

```
problem_43 = undefined
```

## Problem 44

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

Solution:

```
problem_44 = undefined
```

## 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..]]
```

## 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
```

## Problem 47

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

Solution:

```
problem_47 = undefined
```

## 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
```

## Problem 49

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

Solution:

I'm new to haskell, improve here :-)

```
isprime2 n x = if x < n then
if (n `mod` x == 0) then
False
else
isprime2 n (x+1)
else
True
isprime n = isprime2 n 2
quicksort [] = []
quicksort (x:xs) = quicksort [y | y <- xs, y<x ] ++ [x] ++ quicksort [y | y <- xs, y>=x]
-- 'each' works like this: each 1234 => [1,2,3,4]
each n 0 = []
each n len = let x = 10 ^ (len-1)
in n `div` x : each (n `mod` x) (len-1)
ispermut x y = if x /= y then (quicksort (each x 4)) == (quicksort (each y 4))
else False
isin2 a [] = False
isin2 a (b:bs) = if a == b then True else isin2 a bs
isin a [] = False
isin a (b:bs) = if a `isin2` b then True else isin a bs
problem_49_2 prime [] = []
problem_49_2 prime (pr:rest) = if ispermut prime pr then
(pr:(problem_49_2 prime rest))
else
problem_49_2 prime rest
problem_49_1 [] res = res
problem_49_1 (pr:prims) res = if not (pr `isin` res) then
let x = (problem_49_2 pr (pr:prims))
in
if x /= [] then
problem_49_1 prims (res ++ [(pr:x)])
else
problem_49_1 prims res
else
problem_49_1 prims res
problem_49 = problem_49_1 [n | n <- [1000..9999], isprime n] []
```

## Problem 50

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

Solution:

```
problem_50 = undefined
```