# Euler problems/151 to 160

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< Euler problems(Difference between revisions)

(Added problem_152) |
(Fixed several problem, thanks to Buba Smith) |
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Line 40: | Line 40: | ||

11 : dropWhile (< 17) primes ++ [25, 27, 32, 49] |
11 : dropWhile (< 17) primes ++ [25, 27, 32, 49] |
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− | -- All pairs (x, n) where x is a rational number whose reduced |
+ | -- All pairs (x, s) where x is a rational number whose reduced |

-- denominator is not divisible by any prime greater than 3; |
-- denominator is not divisible by any prime greater than 3; |
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− | -- and n>0 is the number of sets of numbers up to 85 divisible |
+ | -- and s is all sets of numbers up to 80 divisible |

-- by a prime greater than 3, whose sum of inverse squares is x. |
-- by a prime greater than 3, whose sum of inverse squares is x. |
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− | only23 = foldl f [(0, 1)] [13, 7, 5] |
+ | only23 = foldl f [(0, [[]])] [13, 7, 5] |

where |
where |
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− | f x p = collect $ concatMap (g p) x |
+ | f a p = collect $ [(y, u ++ v) | (x, s) <- a, |

− | g p (x, n) = map (\(a, b) -> (a, n * length b)) $ pfree (terms p) x p |
+ | (y, v) <- pfree (terms p) x p, |

− | terms p = [n * p | n <- [1..85`div`p], |
+ | u <- s] |

− | all (\q -> n `mod` q /= 0) [5, 7, 11, 13, 17]] |
+ | terms p = [n * p | n <- [1..80`div`p], |

− | collect = map (\z -> (fst $ head z, sum $ map snd z)) |
+ | all (\q -> n `mod` q /= 0) $ |

− | . groupBy cmpFst . sort |
+ | 11 : takeWhile (>= p) [13, 7, 5] |

− | cmpFst x y = fst x == fst y |
+ | ] |

+ | collect = map (\z -> (fst $ head z, map snd z)) . |
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+ | groupBy fstEq . sortBy cmpFst |
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+ | fstEq (x, _) (y, _) = x == y |
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+ | cmpFst (x, _) (y, _) = compare x y |
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-- All subsets (of an ordered set) whose sum of inverse squares is x |
-- All subsets (of an ordered set) whose sum of inverse squares is x |
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Line 64: | Line 64: | ||

f _ _ = [] |
f _ _ = [] |
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− | -- All numbers up to 85 that are divisible only by the primes |
+ | -- All numbers up to 80 that are divisible only by the primes |

-- 2 and 3 and are not divisible by 32 or 27. |
-- 2 and 3 and are not divisible by 32 or 27. |
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− | all23 = [n | a <- [0..4], b <- [0..2], let n = 2^a * 3^b, n <= 85] |
+ | all23 = [n | a <- [0..4], b <- [0..2], let n = 2^a * 3^b, n <= 80] |

− | problem_152 = if verify |
+ | solutions = if verify |

− | then sum [n * length (findInvSq (1%2 - x) all23) | |
+ | then [sort $ u ++ v | (x, s) <- only23, |

− | (x, n) <- only23] |
+ | u <- findInvSq (1%2 - x) all23, |

− | else undefined |
+ | v <- s] |

+ | else undefined |
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+ | |||

+ | problem_152 = length solutions |
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</haskell> |
</haskell> |
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## Revision as of 00:49, 24 September 2007

## Contents |

## 1 Problem 151

Paper sheets of standard sizes: an expected-value problem.

Solution:

problem_151 = undefined

## 2 Problem 152

Writing 1/2 as a sum of inverse squares

Note that if p is an odd prime, the sum of inverse squares of all terms divisible by p must have reduced denominator not divisible by p.

Solution:

import Data.Ratio import Data.List invSq n = 1 % (n * n) sumInvSq = sum . map invSq subsets (x:xs) = let s = subsets xs in s ++ map (x :) s subsets _ = [[]] primes = 2 : 3 : 7 : [p | p <- [11, 13..83], all (\q -> p `mod` q /= 0) [3, 5, 7]] -- All subsets whose sum of inverse squares, -- when added to x, does not contain a factor of p pfree s x p = [(y, t) | t <- subsets s, let y = x + sumInvSq t, denominator y `mod` p /= 0] -- Verify that we need not consider terms divisible by 11, or by any -- prime greater than 13. Nor need we consider any term divisible -- by 25, 27, 32, or 49. verify = all (\p -> null $ tail $ pfree [p, 2*p..85] 0 p) $ 11 : dropWhile (< 17) primes ++ [25, 27, 32, 49] -- All pairs (x, s) where x is a rational number whose reduced -- denominator is not divisible by any prime greater than 3; -- and s is all sets of numbers up to 80 divisible -- by a prime greater than 3, whose sum of inverse squares is x. only23 = foldl f [(0, [[]])] [13, 7, 5] where f a p = collect $ [(y, u ++ v) | (x, s) <- a, (y, v) <- pfree (terms p) x p, u <- s] terms p = [n * p | n <- [1..80`div`p], all (\q -> n `mod` q /= 0) $ 11 : takeWhile (>= p) [13, 7, 5] ] collect = map (\z -> (fst $ head z, map snd z)) . groupBy fstEq . sortBy cmpFst fstEq (x, _) (y, _) = x == y cmpFst (x, _) (y, _) = compare x y -- All subsets (of an ordered set) whose sum of inverse squares is x findInvSq x y = f x $ zip3 y (map invSq y) (map sumInvSq $ init $ tails y) where f 0 _ = [[]] f x ((n, r, s):ns) | r > x = f x ns | s < x = [] | otherwise = map (n :) (f (x - r) ns) ++ f x ns f _ _ = [] -- All numbers up to 80 that are divisible only by the primes -- 2 and 3 and are not divisible by 32 or 27. all23 = [n | a <- [0..4], b <- [0..2], let n = 2^a * 3^b, n <= 80] solutions = if verify then [sort $ u ++ v | (x, s) <- only23, u <- findInvSq (1%2 - x) all23, v <- s] else undefined problem_152 = length solutions

## 3 Problem 153

Investigating Gaussian Integers

Solution:

problem_153 = undefined

## 4 Problem 154

Exploring Pascal's pyramid.

Solution:

problem_154 = undefined

## 5 Problem 155

Counting Capacitor Circuits.

Solution:

problem_155 = undefined

## 6 Problem 156

Counting Digits

Solution:

problem_156 = undefined

## 7 Problem 157

Solving the diophantine equation 1/a+1/b= p/10n

Solution:

problem_157 = undefined

## 8 Problem 158

Exploring strings for which only one character comes lexicographically after its neighbour to the left.

Solution:

problem_158 = undefined

## 9 Problem 159

Digital root sums of factorisations.

Solution:

problem_159 = undefined

## 10 Problem 160

Factorial trailing digits

Solution:

problem_160 = undefined