Euler problems/151 to 160

Problem 151

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

Solution:

problem_151 = undefined

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

Problem 153

Investigating Gaussian Integers

Solution:

problem_153 = undefined

Problem 154

Exploring Pascal's pyramid.

Solution:

problem_154 = undefined

Problem 155

Counting Capacitor Circuits.

Solution:

problem_155 = undefined

Problem 156

Counting Digits

Solution:

problem_156 = undefined

Problem 157

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

Solution:

problem_157 = undefined

Problem 158

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

Solution:

problem_158 = undefined

Problem 159

Digital root sums of factorisations.

Solution:

problem_159 = undefined

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

We really only need these two facts for the special case of d == 5, and we can verify that directly by evaluating the above two Haskell expressions.

More generally:

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)

it have another fast way to do this .

Solution:

import Data.List
mulMod :: Integral a => a -> a -> a -> a
mulMod a b c= (b * c) `rem` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
    where
    f x n y
        | n == 1 = x `mul` y
        | r == 0 = f x2 q y
        | otherwise = f x2 q (x `mul` y)
        where
            (q,r) = quotRem n 2
            x2 = sq x
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)
 
productMod =foldl (mulMod (10^5)) 1
hFacial 0=1
hFacial a
    |gcd a 5==1=mod (a*hFacial(a-1)) (5^5)
    |otherwise=hFacial(a-1)
fastFacial a= hFacial $mod a 6250
numPrime x p=takeWhile(>0) [div x (p^a)|a<-[1..]]
p160 x=mulMod t5 a b
    where
    t5=10^5
    lst=numPrime x 5
    a=powMod t5 1563 $mod c 2500
    b=productMod  c6 
    c=sum lst
    c6=map fastFacial $x:lst
problem_160 = p160 (10^12)