Euler problems/31 to 40

Problem 31

Investigating combinations of English currency denominations.

Solution:

This is the naive doubly recursive solution. Speed would be greatly improved by use of memoization, dynamic programming, or the closed form.

problem_31 = 
    ways [1,2,5,10,20,50,100,200] !!200
    where 
    ways [] = 1 : repeat 0
    ways (coin:coins) =n 
        where
        n = zipWith (+) (ways coins) (take coin (repeat 0) ++ n)

A beautiful solution, making usage of laziness and recursion to implement a dynamic programming scheme, blazingly fast despite actually generating the combinations and not only counting them :

coins = [1,2,5,10,20,50,100,200]

combinations = foldl (\without p ->
                          let (poor,rich) = splitAt p without
                              with = poor ++ 
                                     zipWith (++) (map (map (p:)) with)
                                                  rich
                          in with
                     ) ([[]] : repeat [])

problem_31 = 
    length $ combinations coins !! 200

Problem 32

Find the sum of all numbers that can be written as pandigital products.

Solution:

import Control.Monad
combs 0 xs = [([],xs)]
combs n xs = [(y:ys,rest)|y<-xs, (ys,rest)<-combs (n-1) (delete y xs)]

l2n :: (Integral a) => [a] -> a
l2n = foldl' (\a b -> 10*a+b) 0

swap (a,b) = (b,a)

explode :: (Integral a) => a -> [a]
explode = 
    unfoldr (\a -> if a==0 then Nothing else Just $ swap $ quotRem a 10)

pandigiticals = nub $ do
  (beg,end) <- combs 5 [1..9]
  n <- [1,2]
  let (a,b) = splitAt n beg
      res = l2n a * l2n b
  guard $ sort (explode res) == end
  return res
problem_32 = sum pandigiticals

Problem 33

Discover all the fractions with an unorthodox cancelling method.

Solution:

import Data.Ratio
problem_33 = denominator $product $ rs
{-
 xy/yz = x/z
(10x + y)/(10y+z) = x/z
9xz + yz = 10xy
 -}
rs=[(10*x+y)%(10*y+z) |
    x <- t, 
    y <- t, 
    z <- t,
    x /= y ,
    (9*x*z) + (y*z) == (10*x*y)
    ]
    where
    t=[1..9]

Problem 34

Find the sum of all numbers which are equal to the sum of the factorial of their digits.

Solution:

import Data.Map (fromList ,(!)) 
digits n 
{-  123->[3,2,1]
 -}
    |n<10=[n]
    |otherwise= y:digits x 
    where
    (x,y)=divMod n 10
-- 123 ->321
problem_34 = 
    sum[ x | x <- [3..100000], x == facsum x ]
    where
    fact n = product [1..n]
    fac=fromList [(a,fact a)|a<-[0..9]]
    facsum x= sum [fac!a|a<-digits x]

Problem 35

How many circular primes are there below one million?

Solution: millerRabinPrimality on the Prime_numbers page

isPrime x
    |x==1=False
    |x==2=True
    |x==3=True
    |otherwise=millerRabinPrimality x 2
permutations n = 
    take l $ map (read . take l) $ 
    tails $ take (2*l -1) $ cycle s
    where
    s = show n
    l = length s
circular_primes []     = []
circular_primes (x:xs)
    | all isPrime p = x :  circular_primes xs
    | otherwise     = circular_primes xs
    where
    p = permutations x
x=[1,3,7,9] 
dmm=(\x y->x*10+y)
x3=[foldl dmm 0 [a,b,c]|a<-x,b<-x,c<-x]
x4=[foldl dmm 0 [a,b,c,d]|a<-x,b<-x,c<-x,d<-x]
x5=[foldl dmm 0 [a,b,c,d,e]|a<-x,b<-x,c<-x,d<-x,e<-x]
x6=[foldl dmm 0 [a,b,c,d,e,f]|a<-x,b<-x,c<-x,d<-x,e<-x,f<-x]
problem_35 = 
    (+13)$length $ circular_primes $ [a|a<-foldl (++) [] [x3,x4,x5,x6],isPrime a]

Problem 36

Find the sum of all numbers less than one million, which are palindromic in base 10 and base 2.

Solution:

isPalin [] = True
isPalin [a] = True
isPalin (x:xs) = 
    if x == last xs then isPalin $ sansLast xs else False
	where 
    sansLast xs = reverse $ tail $ reverse xs
toBase2 0 = []
toBase2 x = (show $ mod x 2) : toBase2 (div x 2)
isbothPalin x = 
    isPalin (show x) && isPalin (toBase2 x)
problem_36= 
    sum $ filter isbothPalin $ filter (not.even) [1..1000000]

Problem 37

Find the sum of all eleven primes that are both truncatable from left to right and right to left.

Solution:

-- isPrime in p35
clist n = 
    filter isLeftTruncatable $ if isPrime n then n:ns else []
    where
    ns = concatMap (clist . ((10*n) +)) [1,3,7,9]
 
isLeftTruncatable =
    all isPrime . map read . init . tail . tails . show
problem_37 =
    sum $ filter (>=10) $ concatMap clist [2,3,5,7]

Problem 38

What is the largest 1 to 9 pandigital that can be formed by multiplying a fixed number by 1, 2, 3, ... ?

Solution:

import Data.List

mult n i vs 
    | length (concat vs) >= 9 = concat vs
    | otherwise               = mult n (i+1) (vs ++ [show (n * i)])

problem_38 = 
    maximum $ map read $ filter
    ((['1'..'9'] ==) .sort) $
    [ mult n 1 [] | n <- [2..9999] ]

Problem 39

If p is the perimeter of a right angle triangle, {a, b, c}, which value, for p ≤ 1000, has the most solutions?

Solution: We use the well known formula to generate primitive Pythagorean triples. All we need are the perimeters, and they have to be scaled to produce all triples in the problem space.

problem_39 = 
    head $ perims !! indexMax
    where
    perims = group $ sort [n*p | p <- pTriples, n <- [1..1000 `div` p]]
    counts = map length perims
    Just indexMax = findIndex (== (maximum counts)) $ counts
    pTriples = 
        [p |
        n <- [1..floor (sqrt 1000)],
        m <- [n+1..floor (sqrt 1000)],
        even n || even m,
        gcd n m == 1,
        let a = m^2 - n^2,
        let b = 2*m*n,
        let c = m^2 + n^2,
        let p = a + b + c,
        p < 1000
        ]

Problem 40

Finding the nth digit of the fractional part of the irrational number.

Solution:

takeLots :: [Int] -> [a] -> [a]
takeLots = 
    t 1 
    where
    t  i [] _  = []
    t  i jj@(j:js) (x:xs) 
        | i == j    = x : t (i+1) js xs
        | otherwise =     t (i+1) jj xs
 
digitos :: [Int]
digitos =
    d [1]
    where
    d k = reverse k ++ d (mais k)
    mais (9:is) = 0 : mais is
    mais (i:is) = (i+1) : is
    mais []     = [1]
 
problem_40 =
    product $ takeLots [10^n | n <- [0..6]] digitos