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

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

Evaluate the sum of all amicable pairs under 10000. |
Evaluate the sum of all amicable pairs under 10000. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=22 Problem 22] == |

What is the total of all the name scores in the file of first names? |
What is the total of all the name scores in the file of first names? |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=23 Problem 23] == |

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers. |
Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=24 Problem 24] == |

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9? |
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9? |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=25 Problem 25] == |

What is the first term in the Fibonacci sequence to contain 1000 digits? |
What is the first term in the Fibonacci sequence to contain 1000 digits? |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=26 Problem 26] == |

Find the value of d < 1000 for which 1/d contains the longest recurring cycle. |
Find the value of d < 1000 for which 1/d contains the longest recurring cycle. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=27 Problem 27] == |

Find a quadratic formula that produces the maximum number of primes for consecutive values of n. |
Find a quadratic formula that produces the maximum number of primes for consecutive values of n. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=28 Problem 28] == |

What is the sum of both diagonals in a 1001 by 1001 spiral? |
What is the sum of both diagonals in a 1001 by 1001 spiral? |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=29 Problem 29] == |

How many distinct terms are in the sequence generated by a<sup>b</sup> for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100? |
How many distinct terms are in the sequence generated by a<sup>b</sup> for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100? |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=30 Problem 30] == |

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits. |
Find the sum of all the numbers that can be written as the sum of fifth powers of their digits. |
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## Revision as of 13:46, 22 January 2008

## Contents

## Problem 21

Evaluate the sum of all amicable pairs under 10000.

Solution:

```
problem_21 =
sum [n |
n <- [2..9999],
let m = eulerTotient n,
m > 1,
m < 10000,
n == eulerTotient m
]
```

## 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 $ show $ 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:

```
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 = putStrLn $ show $ foldl1 (+) $ 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
```

## Problem 25

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

Solution:

```
import Data.List
fib x
|x==0=0
|x==1=1
|x==2=1
|odd x=(fib (d+1))^2+(fib d)^2
|otherwise=(fib (d+1))^2-(fib (d-1))^2
where
d=div x 2
phi=(1+sqrt 5)/2
dig x=floor( (fromInteger x-1) * log 10 /log phi)
problem_25 =
head[a|a<-[dig num..],(>=limit)$fib a]
where
num=1000
limit=10^(num-1)
```

## Problem 26

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

Solution:

```
next n d = (n `mod` d):next (10*n`mod`d) d
idigs n = tail $ take (1+n) $ next 1 n
pos x = map fst . filter ((==x) . snd) . zip [1..]
periods n = let d = idigs n in pos (head d) (tail d)
problem_26 =
snd$maximum [(m,a)|
a<-[800..1000] ,
let k=periods a,
not$null k,
let m=head k
]
```

## Problem 27

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

Solution:

```
eulerCoefficients n
= [((len, a*b), (a, b))
| b <- takeWhile (<n) primes, a <- [-b+1..n-1],
let len = length $ takeWhile (isPrime . (\x -> x^2 + a*x + b)) [0..],
if b == 2 then even a else odd a, len > 39]
problem_27 = snd . fst . maximum . eulerCoefficients $ 1000
```

## 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
```

## 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]
```

## 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.Array
import Data.Char
p = listArray (0,9) $ map (^5) [0..9]
upperLimit = 295277
candidates =
[ n |
n <- [10..upperLimit],
(sum $ digits n) `mod` 10 == last(digits n),
powersum n == n
]
where
digits n = map digitToInt $ show n
powersum n = sum $ map (p!) $ digits n
problem_30 = sum candidates
```