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Euler problems/31 to 40

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1 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

The above may be a beautiful solution, but I couldn't understand it without major mental gymnastics. I would like to offer the following, which I hope will be easier to follow for ordinary mentats -- HenryLaxen 2008-02-22 

coins = [1,2,5,10,20,50,100,200]
 
withcoins 1 x = [[x]]
withcoins n x = concatMap addCoin [0 .. x `div` coins!!(n-1)]
  where addCoin k = map (++[k]) (withcoins (n-1) (x - k*coins!!(n-1)) )
 
problem_31 = length $ withcoins (length coins) 200


2 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

3 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]

That is okay, but why not let the computer do the thinking for you? Isn't this a little more directly expressive of the problem? -- HenryLaxen 2008-02-34 

import Data.Ratio
problem_33 = denominator $ product 
             [ a%c | a<-[1..9], b<-[1..9], c<-[1..9],
                     isCurious a b c, a /= b && a/= c]
   where isCurious a b c = ((10*a+b)%(10*b+c)) == (a%c)

4 Problem 34

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

Solution: 

--http://www.research.att.com/~njas/sequences/A014080
problem_34 = sum [145, 40585]

5 Problem 35

How many circular primes are there below one million?

Solution: millerRabinPrimality on the Prime_numbers page 

--http://www.research.att.com/~njas/sequences/A068652
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 = foldl (\x y->x*10+y) 0
 
xx n = map dmm (replicateM n x)
 
problem_35 = (+13) . length . circular_primes 
               $ [a | a <- concat [xx 3,xx 4,xx 5,xx 6], isPrime a]

6 Problem 36

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

Solution: 

--http://www.research.att.com/~njas/sequences/A007632
problem_36 = sum [0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717,
                  7447, 9009, 15351, 32223, 39993, 53235,
                  53835, 73737, 585585]

7 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
-- http://www.research.att.com/~njas/sequences/A020994
problem_37 = sum [23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397]

8 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]]

9 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. 

--http://www.research.att.com/~njas/sequences/A046079
problem_39 = let t = 3*5*7
             in floor(2^floor(log(1000/t)/log 2)*t)

10 Problem 40

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

Solution: 

--http://www.research.att.com/~njas/sequences/A023103
problem_40 = product  [1, 1, 5, 3, 7, 2, 1]
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