# Euler problems/51 to 60

### From HaskellWiki

(→[http://projecteuler.net/index.php?section=view&id=54 Problem 54]) |
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Solution: |
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<haskell> |
<haskell> |
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− | problem_58 = undefined |
+ | base :: (Integral a) => [a] |

+ | base = base' 2 |
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+ | where |
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+ | base' n = n:n:n:n:(base' $ n + 2) |
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+ | |||

+ | pascal = scanl (+) 1 base |
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+ | |||

+ | ratios :: [Integer] -> [Double] |
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+ | ratios (x:xs) = 1.0 : ratios' 0 1 xs |
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+ | where |
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+ | ratios' n d (w:x:y:z:xs) = ((fromInteger num)/(fromInteger den)) : (ratios' num den xs) |
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+ | where |
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+ | num = (p w + p x + p y + p z + n) |
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+ | den = (d + 4) |
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+ | p n = case isPrime n of |
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+ | True -> 1 |
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+ | False -> 0 |
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+ | |||

+ | problem_58 = fst $ head $ dropWhile (\(_,a) -> a > 0.1) $ zip [1,3..] (ratios pascal) |
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</haskell> |
</haskell> |
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## Revision as of 14:46, 8 August 2007

## Contents |

## 1 Problem 51

Find the smallest prime which, by changing the same part of the number, can form eight different primes.

Solution:

problem_51 = undefined

## 2 Problem 52

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.

Solution:

problem_52 = head [n | n <- [1..], digits (2*n) == digits (3*n), digits (3*n) == digits (4*n), digits (4*n) == digits (5*n), digits (5*n) == digits (6*n)] where digits = sort . show

## 3 Problem 53

How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?

Solution:

problem_53 = length [n | n <- [1..100], r <- [1..n], n `choose` r > 10^6] where n `choose` r | r > n || r < 0 = 0 | otherwise = foldl (\z j -> z*(n-j+1) `div` j) n [2..r]

## 4 Problem 54

How many hands did player one win in the poker games?

Solution:

problem_54 = undefined

## 5 Problem 55

How many Lychrel numbers are there below ten-thousand?

Solution:

problem_55 = length $ filter isLychrel [1..9999] where isLychrel n = all notPalindrome (take 50 (tail (iterate revadd n))) notPalindrome s = (show s) /= reverse (show s) revadd n = n + rev n where rev n = read (reverse (show n))

## 6 Problem 56

Considering natural numbers of the form, a^{b}, finding the maximum digital sum.

Solution:

problem_56 = maximum [dsum (a^b) | a <- [1..99], b <-[1..99]] where dsum 0 = 0 dsum n = let ( d, m ) = n `divMod` 10 in m + ( dsum d )

## 7 Problem 57

Investigate the expansion of the continued fraction for the square root of two.

Solution:

problem_57 = length $ filter topHeavy $ take 1000 convergents where topHeavy r = numDigits (numerator r) > numDigits (denominator r) numDigits = length . show convergents = iterate next (3%2) next r = 1 + 1/(1+r)

## 8 Problem 58

Investigate the number of primes that lie on the diagonals of the spiral grid.

Solution:

base :: (Integral a) => [a] base = base' 2 where base' n = n:n:n:n:(base' $ n + 2) pascal = scanl (+) 1 base ratios :: [Integer] -> [Double] ratios (x:xs) = 1.0 : ratios' 0 1 xs where ratios' n d (w:x:y:z:xs) = ((fromInteger num)/(fromInteger den)) : (ratios' num den xs) where num = (p w + p x + p y + p z + n) den = (d + 4) p n = case isPrime n of True -> 1 False -> 0 problem_58 = fst $ head $ dropWhile (\(_,a) -> a > 0.1) $ zip [1,3..] (ratios pascal)

## 9 Problem 59

Using a brute force attack, can you decrypt the cipher using XOR encryption?

Solution:

problem_59 = undefined

## 10 Problem 60

Find a set of five primes for which any two primes concatenate to produce another prime.

Solution:

problem_60 = undefined