# Euler problems/151 to 160

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## 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..79], 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

We use the following two facts:

Fact 1: (2^(d + 4*5^(d-1)) - 2^d) `mod` 10^d == 0

Fact 2: product [n | n <- [0..10^d], gcd n 10 == 1] `mod` 10^d == 1

Fact 1 follows from the fact that the group of invertible elements of the ring of integers modulo 5^d has 4*5^(d-1) elements.

Fact 2 follows from the fact that the group of invertible elements of the ring of integers modulo 10^d is isomorphic to the product of a cyclic group of order 2 and another cyclic group.

Solution:

problem_160 = trailingFactorialDigits 5 (10^12) trailingFactorialDigits d n = twos `times` odds where base = 10 ^ d x `times` y = (x * y) `mod` base multiply = foldl' times 1 x `toPower` k = multiply $ genericReplicate n x e = facFactors 2 n - facFactors 5 n twos | e <= d = 2 `toPower` e | otherwise = 2 `toPower` (d + (e - d) `mod` (4 * 5 ^ (d - 1))) odds = multiply [odd | a <- takeWhile (<= n) $ iterate (* 2) 1, b <- takeWhile (<= n) $ iterate (* 5) a, odd <- [3, 5 .. n `div` b `mod` base], odd `mod` 5 /= 0] -- The number of factors of the prime p in n! facFactors p = sum . zipWith (*) (iterate (\x -> p * x + 1) 1) . tail . radix p -- The digits of n in base b representation radix p = map snd . takeWhile (/= (0, 0)) . iterate ((`divMod` p) . fst) . (`divMod` p)