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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

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]

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=(\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]

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]