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Euler problems/151 to 160

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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` n = multiply . genericTake n $ 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)