# Difference between revisions of "Euler problems/21 to 30"

(→[http://projecteuler.net/index.php?section=problems&id=29 Problem 29]: a solution) |
(→[http://projecteuler.net/index.php?section=problems&id=21 Problem 21]: a solution) |
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Line 3: | Line 3: | ||

Solution: |
Solution: |
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+ | This is a little slow because of the naive method used to compute the divisors. |
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<haskell> |
<haskell> |
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− | problem_21 = undefined |
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+ | problem_21 = sum [m+n | m <- [2..9999], let n = divisorsSum ! m, amicable m n] |
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+ | where amicable m n = m < n && n < 10000 && divisorsSum ! n == m |
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+ | divisorsSum = array (1,9999) |
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+ | [(i, sum (divisors i)) | i <- [1..9999]] |
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+ | divisors n = [j | j <- [1..n `div` 2], n `mod` j == 0] |
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</haskell> |
</haskell> |
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## Revision as of 07:19, 30 March 2007

## Contents

## Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution: This is a little slow because of the naive method used to compute the divisors.

```
problem_21 = sum [m+n | m <- [2..9999], let n = divisorsSum ! m, amicable m n]
where amicable m n = m < n && n < 10000 && divisorsSum ! n == m
divisorsSum = array (1,9999)
[(i, sum (divisors i)) | i <- [1..9999]]
divisors n = [j | j <- [1..n `div` 2], n `mod` j == 0]
```

## Problem 22

What is the total of all the name scores in the file of first names?

Solution:

```
-- apply to a list of names
problem_22 :: [String] -> Int
problem_22 = sum . zipWith (*) [ 1 .. ] . map score
where score = sum . map ( subtract 64 . ord )
```

## Problem 23

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

Solution:

```
problem_23 = undefined
```

## Problem 24

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

Solution:

```
perms [] = [[]]
perms xs = do
x <- xs
map ( x: ) ( perms . delete x $ xs )
problem_24 = ( perms "0123456789" ) !! 999999
```

## Problem 25

What is the first term in the Fibonacci sequence to contain 1000 digits?

Solution:

```
valid ( i, n ) = length ( show n ) == 1000
problem_25 = fst . head . filter valid . zip [ 1 .. ] $ fibs
where fibs = 1 : 1 : 2 : zipWith (+) fibs ( tail fibs )
```

## Problem 26

Find the value of d < 1000 for which 1/d contains the longest recurring cycle.

Solution:

```
problem_26 = undefined
```

## Problem 27

Find a quadratic formula that produces the maximum number of primes for consecutive values of n.

Solution:

```
problem_27 = undefined
```

## Problem 28

What is the sum of both diagonals in a 1001 by 1001 spiral?

Solution:

```
problem_28 = undefined
```

## Problem 29

How many distinct terms are in the sequence generated by a^{b} for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?

Solution:

```
problem_29 = length . group . sort $ [a^b | a <- [2..100], b <- [2..100]]
```

## Problem 30

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

Solution:

```
problem_30 = undefined
```