# Difference between revisions of "Euler problems/41 to 50"

m (→[http://projecteuler.net/index.php?section=problems&id=48 Problem 48]) |
(→[http://projecteuler.net/index.php?section=problems&id=41 Problem 41]: a solution) |
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Line 4: | Line 4: | ||

Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_41 = undefined |
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+ | problem_41 = head [p | n <- init (tails "987654321"), |
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+ | p <- perms n, isPrime (read p)] |
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+ | where perms [] = [[]] |
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+ | perms xs = [x:ps | x <- xs, ps <- perms (delete x xs)] |
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+ | isPrime n = n > 1 && smallestDivisor n == n |
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+ | smallestDivisor n = findDivisor n (2:[3,5..]) |
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+ | findDivisor n (testDivisor:rest) |
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+ | | n `mod` testDivisor == 0 = testDivisor |
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+ | | testDivisor*testDivisor >= n = n |
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+ | | otherwise = findDivisor n rest |
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</haskell> |
</haskell> |
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## Revision as of 01:25, 11 April 2007

## 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:

```
problem_42 = undefined
```

## 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:

```
problem_49 = undefined
```

## Problem 50

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

Solution:

```
problem_50 = undefined
```