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

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== [http://projecteuler.net/index.php?section=problems&id=21 Problem 21] == | == [http://projecteuler.net/index.php?section=problems&id=21 Problem 21] == | ||

− | Evaluate the sum of all amicable | + | Evaluate the sum of all amicable numbers (including those with a pair number over the limit) under 10000. |

+ | |||

+ | Solution: | ||

+ | (http://www.research.att.com/~njas/sequences/A063990) | ||

+ | |||

+ | This is a little slow because of the naive method used to compute the divisors. | ||

+ | <haskell> | ||

+ | 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] | ||

+ | </haskell> | ||

+ | |||

+ | Here is an alternative using a faster way of computing the sum of divisors. | ||

+ | <haskell> | ||

+ | problem_21_v2 = sum [n | n <- [2..9999], let m = d n, | ||

+ | m > 1, m < 10000, n == d m, d m /= d (d m)] | ||

+ | d n = product [(p * product g - 1) `div` (p - 1) | | ||

+ | g <- group $ primeFactors n, let p = head g | ||

+ | ] - n | ||

+ | primeFactors = pf primes | ||

+ | where | ||

+ | pf ps@(p:ps') n | ||

+ | | p * p > n = [n] | ||

+ | | r == 0 = p : pf ps q | ||

+ | | otherwise = pf ps' n | ||

+ | where (q, r) = n `divMod` p | ||

+ | primes = 2 : filter (null . tail . primeFactors) [3,5..] | ||

+ | </haskell> | ||

+ | |||

+ | Here is another alternative solution that computes the sum-of-divisors for the numbers by iterating over products of their factors (very fast): | ||

− | |||

<haskell> | <haskell> | ||

− | + | import Data.Array | |

+ | |||

+ | max_ = 100000 | ||

+ | |||

+ | gen 100001 = [] | ||

+ | gen n = [(i*n,n)|i <- [2 .. max_ `div` n]] ++ (gen (n+1)) | ||

+ | |||

+ | arr = accumArray (+) 0 (0,max_) (gen 1) | ||

+ | |||

+ | problem_21_v3 = sum $ filter (\a -> let b = (arr!a) in b /= a && (arr!b) == a) [1 .. (10000 - 1)] | ||

+ | |||

</haskell> | </haskell> | ||

Line 12: | Line 52: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | - | + | import Data.List |

− | + | import Data.Char | |

− | + | problem_22 = | |

− | + | do input <- readFile "names.txt" | |

+ | let names = sort $ read$"["++ input++"]" | ||

+ | let scores = zipWith score names [1..] | ||

+ | print . sum $ scores | ||

+ | where score w i = (i *) . sum . map (\c -> ord c - ord 'A' + 1) $ w | ||

</haskell> | </haskell> | ||

Line 23: | Line 67: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | problem_23 = | + | --http://www.research.att.com/~njas/sequences/A048242 |

+ | import Data.Array | ||

+ | n = 28124 | ||

+ | abundant n = eulerTotient n - n > n | ||

+ | abunds_array = listArray (1,n) $ map abundant [1..n] | ||

+ | abunds = filter (abunds_array !) [1..n] | ||

+ | |||

+ | rests x = map (x-) $ takeWhile (<= x `div` 2) abunds | ||

+ | isSum = any (abunds_array !) . rests | ||

+ | |||

+ | problem_23 = print . sum . filter (not . isSum) $ [1..n] | ||

</haskell> | </haskell> | ||

Line 31: | Line 85: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | perms [] = [ | + | import Data.List |

− | perms xs = | + | |

− | + | fac 0 = 1 | |

− | + | fac n = n * fac (n - 1) | |

+ | perms [] _= [] | ||

+ | perms xs n= x : perms (delete x xs) (mod n m) | ||

+ | where m = fac $ length xs - 1 | ||

+ | y = div n m | ||

+ | x = xs!!y | ||

+ | |||

+ | problem_24 = perms "0123456789" 999999 | ||

+ | </haskell> | ||

+ | |||

+ | Or, using Data.List.permutations, | ||

+ | <haskell> | ||

+ | import Data.List | ||

+ | problem_24 = (!! 999999) . sort $ permutations ['0'..'9'] | ||

+ | </haskell> | ||

+ | |||

+ | Casey Hawthorne | ||

+ | |||

+ | For Project Euler #24 you don't need to generate all the lexicographic permutations by Knuth's method or any other. | ||

+ | |||

+ | You're only looking for the millionth lexicographic permutation of "0123456789" | ||

+ | |||

+ | <haskell> | ||

+ | |||

+ | -- Plan of attack. | ||

+ | |||

+ | -- The "x"s are different numbers | ||

+ | -- 0xxxxxxxxx represents 9! = 362880 permutations/numbers | ||

+ | -- 1xxxxxxxxx represents 9! = 362880 permutations/numbers | ||

+ | -- 2xxxxxxxxx represents 9! = 362880 permutations/numbers | ||

+ | |||

+ | |||

+ | -- 20xxxxxxxx represents 8! = 40320 | ||

+ | -- 21xxxxxxxx represents 8! = 40320 | ||

+ | |||

+ | -- 23xxxxxxxx represents 8! = 40320 | ||

+ | -- 24xxxxxxxx represents 8! = 40320 | ||

+ | -- 25xxxxxxxx represents 8! = 40320 | ||

+ | -- 26xxxxxxxx represents 8! = 40320 | ||

+ | -- 27xxxxxxxx represents 8! = 40320 | ||

+ | |||

+ | |||

+ | module Euler where | ||

+ | |||

+ | import Data.List | ||

+ | |||

+ | factorial n = product [1..n] | ||

+ | |||

+ | -- lexOrder "0123456789" 1000000 "" | ||

+ | |||

+ | lexOrder digits left s | ||

+ | | len == 0 = s ++ digits | ||

+ | | quot > 0 && rem == 0 = lexOrder (digits\\(show (digits!!(quot-1)))) rem (s ++ [(digits!!(quot-1))]) | ||

+ | | quot == 0 && rem == 0 = lexOrder (digits\\(show (digits!!len))) rem (s ++ [(digits!!len)]) | ||

+ | | rem == 0 = lexOrder (digits\\(show (digits!!(quot+1)))) rem (s ++ [(digits!!(quot+1))]) | ||

+ | | otherwise = lexOrder (digits\\(show (digits!!(quot)))) rem (s ++ [(digits!!(quot))]) | ||

+ | where | ||

+ | len = (length digits) - 1 | ||

+ | (quot,rem) = quotRem left (factorial len) | ||

− | |||

</haskell> | </haskell> | ||

Line 44: | Line 155: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | + | fibs = 0:1:(zipWith (+) fibs (tail fibs)) | |

+ | t = 10^999 | ||

+ | |||

+ | problem_25 = length w | ||

+ | where | ||

+ | w = takeWhile (< t) fibs | ||

+ | </haskell> | ||

+ | |||

+ | |||

+ | Casey Hawthorne | ||

+ | |||

+ | I believe you mean the following: | ||

+ | |||

+ | <haskell> | ||

+ | |||

+ | fibs = 0:1:(zipWith (+) fibs (tail fibs)) | ||

− | + | last (takeWhile (<10^1000) fibs) | |

− | |||

</haskell> | </haskell> | ||

Line 55: | Line 180: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | problem_26 = | + | problem_26 = fst $ maximumBy (comparing snd) |

+ | [(n,recurringCycle n) | n <- [1..999]] | ||

+ | where recurringCycle d = remainders d 10 [] | ||

+ | remainders d 0 rs = 0 | ||

+ | remainders d r rs = let r' = r `mod` d | ||

+ | in case elemIndex r' rs of | ||

+ | Just i -> i + 1 | ||

+ | Nothing -> remainders d (10*r') (r':rs) | ||

</haskell> | </haskell> | ||

Line 63: | Line 195: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | problem_27 = | + | problem_27 = -(2*a-1)*(a^2-a+41) |

+ | where n = 1000 | ||

+ | m = head $ filter (\x->x^2-x+41>n) [1..] | ||

+ | a = m-1 | ||

</haskell> | </haskell> | ||

Line 71: | Line 206: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | problem_28 = | + | problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1 |

+ | </haskell> | ||

+ | |||

+ | Alternatively, one can use the fact that the distance between the diagonal numbers increases by 2 in every concentric square. Each square contains four gaps, so the following <hask>scanl</hask> does the trick: | ||

+ | |||

+ | <haskell> | ||

+ | euler28 n = sum $ scanl (+) 0 | ||

+ | (1:(concatMap (replicate 4) [2,4..(n-1)])) | ||

</haskell> | </haskell> | ||

== [http://projecteuler.net/index.php?section=problems&id=29 Problem 29] == | == [http://projecteuler.net/index.php?section=problems&id=29 Problem 29] == | ||

− | How many distinct terms are in the sequence generated by | + | How many distinct terms are in the sequence generated by a<sup>b</sup> for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100? |

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | problem_29 = | + | import Control.Monad |

+ | problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100] | ||

+ | </haskell> | ||

+ | |||

+ | We can also solve it in a more naive way, without using Monads, like this: | ||

+ | <haskell> | ||

+ | import List | ||

+ | problem_29 = length $ nub pr29_help | ||

+ | where pr29_help = [z | y <- [2..100], | ||

+ | z <- lift y] | ||

+ | lift y = map (\x -> x^y) [2..100] | ||

+ | </haskell> | ||

+ | |||

+ | Simpler: | ||

+ | |||

+ | <haskell> | ||

+ | import List | ||

+ | problem_29 = length $ nub [x^y | x <- [2..100], y <- [2..100]] | ||

+ | </haskell> | ||

+ | |||

+ | Instead of using lists, the Set data structure can be used for a significant speed increase: | ||

+ | |||

+ | <haskell> | ||

+ | import Set | ||

+ | problem_29 = size $ fromList [x^y | x <- [2..100], y <- [2..100]] | ||

</haskell> | </haskell> | ||

Line 87: | Line 253: | ||

Solution: | Solution: | ||

<haskell> | <haskell> | ||

− | + | import Data.Char (digitToInt) | |

− | + | ||

+ | limit :: Integer | ||

+ | limit = snd $ head $ dropWhile (\(a,b) -> a > b) $ zip (map (9^5*) [1..]) (map (10^) [1..]) | ||

+ | fifth :: Integer -> Integer | ||

+ | fifth = sum . map ((^5) . toInteger . digitToInt) . show | ||

− | + | problem_30 :: Integer | |

− | [ | + | problem_30 = sum $ filter (\n -> n == fifth n) [2..limit] |

+ | </haskell> |

## Latest revision as of 15:53, 11 October 2015

## Contents

## Problem 21

Evaluate the sum of all amicable numbers (including those with a pair number over the limit) under 10000.

Solution: (http://www.research.att.com/~njas/sequences/A063990)

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]
```

Here is an alternative using a faster way of computing the sum of divisors.

```
problem_21_v2 = sum [n | n <- [2..9999], let m = d n,
m > 1, m < 10000, n == d m, d m /= d (d m)]
d n = product [(p * product g - 1) `div` (p - 1) |
g <- group $ primeFactors n, let p = head g
] - n
primeFactors = pf primes
where
pf ps@(p:ps') n
| p * p > n = [n]
| r == 0 = p : pf ps q
| otherwise = pf ps' n
where (q, r) = n `divMod` p
primes = 2 : filter (null . tail . primeFactors) [3,5..]
```

Here is another alternative solution that computes the sum-of-divisors for the numbers by iterating over products of their factors (very fast):

```
import Data.Array
max_ = 100000
gen 100001 = []
gen n = [(i*n,n)|i <- [2 .. max_ `div` n]] ++ (gen (n+1))
arr = accumArray (+) 0 (0,max_) (gen 1)
problem_21_v3 = sum $ filter (\a -> let b = (arr!a) in b /= a && (arr!b) == a) [1 .. (10000 - 1)]
```

## Problem 22

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

Solution:

```
import Data.List
import Data.Char
problem_22 =
do input <- readFile "names.txt"
let names = sort $ read$"["++ input++"]"
let scores = zipWith score names [1..]
print . sum $ scores
where score w i = (i *) . sum . map (\c -> ord c - ord 'A' + 1) $ w
```

## Problem 23

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

Solution:

```
--http://www.research.att.com/~njas/sequences/A048242
import Data.Array
n = 28124
abundant n = eulerTotient n - n > n
abunds_array = listArray (1,n) $ map abundant [1..n]
abunds = filter (abunds_array !) [1..n]
rests x = map (x-) $ takeWhile (<= x `div` 2) abunds
isSum = any (abunds_array !) . rests
problem_23 = print . sum . filter (not . isSum) $ [1..n]
```

## Problem 24

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

Solution:

```
import Data.List
fac 0 = 1
fac n = n * fac (n - 1)
perms [] _= []
perms xs n= x : perms (delete x xs) (mod n m)
where m = fac $ length xs - 1
y = div n m
x = xs!!y
problem_24 = perms "0123456789" 999999
```

Or, using Data.List.permutations,

```
import Data.List
problem_24 = (!! 999999) . sort $ permutations ['0'..'9']
```

Casey Hawthorne

For Project Euler #24 you don't need to generate all the lexicographic permutations by Knuth's method or any other.

You're only looking for the millionth lexicographic permutation of "0123456789"

```
-- Plan of attack.
-- The "x"s are different numbers
-- 0xxxxxxxxx represents 9! = 362880 permutations/numbers
-- 1xxxxxxxxx represents 9! = 362880 permutations/numbers
-- 2xxxxxxxxx represents 9! = 362880 permutations/numbers
-- 20xxxxxxxx represents 8! = 40320
-- 21xxxxxxxx represents 8! = 40320
-- 23xxxxxxxx represents 8! = 40320
-- 24xxxxxxxx represents 8! = 40320
-- 25xxxxxxxx represents 8! = 40320
-- 26xxxxxxxx represents 8! = 40320
-- 27xxxxxxxx represents 8! = 40320
module Euler where
import Data.List
factorial n = product [1..n]
-- lexOrder "0123456789" 1000000 ""
lexOrder digits left s
| len == 0 = s ++ digits
| quot > 0 && rem == 0 = lexOrder (digits\\(show (digits!!(quot-1)))) rem (s ++ [(digits!!(quot-1))])
| quot == 0 && rem == 0 = lexOrder (digits\\(show (digits!!len))) rem (s ++ [(digits!!len)])
| rem == 0 = lexOrder (digits\\(show (digits!!(quot+1)))) rem (s ++ [(digits!!(quot+1))])
| otherwise = lexOrder (digits\\(show (digits!!(quot)))) rem (s ++ [(digits!!(quot))])
where
len = (length digits) - 1
(quot,rem) = quotRem left (factorial len)
```

## Problem 25

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

Solution:

```
fibs = 0:1:(zipWith (+) fibs (tail fibs))
t = 10^999
problem_25 = length w
where
w = takeWhile (< t) fibs
```

Casey Hawthorne

I believe you mean the following:

```
fibs = 0:1:(zipWith (+) fibs (tail fibs))
last (takeWhile (<10^1000) fibs)
```

## Problem 26

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

Solution:

```
problem_26 = fst $ maximumBy (comparing snd)
[(n,recurringCycle n) | n <- [1..999]]
where recurringCycle d = remainders d 10 []
remainders d 0 rs = 0
remainders d r rs = let r' = r `mod` d
in case elemIndex r' rs of
Just i -> i + 1
Nothing -> remainders d (10*r') (r':rs)
```

## Problem 27

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

Solution:

```
problem_27 = -(2*a-1)*(a^2-a+41)
where n = 1000
m = head $ filter (\x->x^2-x+41>n) [1..]
a = m-1
```

## Problem 28

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

Solution:

```
problem_28 = sum (map (\n -> 4*(n-2)^2+10*(n-1)) [3,5..1001]) + 1
```

Alternatively, one can use the fact that the distance between the diagonal numbers increases by 2 in every concentric square. Each square contains four gaps, so the following `scanl`

does the trick:

```
euler28 n = sum $ scanl (+) 0
(1:(concatMap (replicate 4) [2,4..(n-1)]))
```

## Problem 29

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

Solution:

```
import Control.Monad
problem_29 = length . group . sort $ liftM2 (^) [2..100] [2..100]
```

We can also solve it in a more naive way, without using Monads, like this:

```
import List
problem_29 = length $ nub pr29_help
where pr29_help = [z | y <- [2..100],
z <- lift y]
lift y = map (\x -> x^y) [2..100]
```

Simpler:

```
import List
problem_29 = length $ nub [x^y | x <- [2..100], y <- [2..100]]
```

Instead of using lists, the Set data structure can be used for a significant speed increase:

```
import Set
problem_29 = size $ fromList [x^y | x <- [2..100], y <- [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:

```
import Data.Char (digitToInt)
limit :: Integer
limit = snd $ head $ dropWhile (\(a,b) -> a > b) $ zip (map (9^5*) [1..]) (map (10^) [1..])
fifth :: Integer -> Integer
fifth = sum . map ((^5) . toInteger . digitToInt) . show
problem_30 :: Integer
problem_30 = sum $ filter (\n -> n == fifth n) [2..limit]
```