# Euler problems/31 to 40

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## Revision as of 12:09, 30 September 2007

## Contents |

## 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 = pence 200 [1,2,5,10,20,50,100,200] where pence 0 _ = 1 pence n [] = 0 pence n denominations@(d:ds) | n < d = 0 | otherwise = pence (n - d) denominations + pence n ds

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:

problem_32 = sum $ nub $ map (\(a, b) -> a * b) multiplicands where multiplicands = [(a,b)| a <- [2..5000], b <- [a..(9999 `div` a)], check a b] check a b = no_zero s && (length ss) == 9 && foldr (\x y -> length x == 1 && y) True ss where s = show a ++ show b ++ show (a*b) ss = group $ sort s no_zero (x:xs) | x == '0' = False | null xs = True | otherwise = no_zero xs

## 3 Problem 33

Discover all the fractions with an unorthodox cancelling method.

Solution:

import Ratio problem_33 = denominator (product $ rs ++ rs') rs = [(x%y) | a <- [0..9], b <- [1..9], c <- [1..9], let x = 10*a + c, let y = 10*c + b, x /= y, x%y < 1, x%y == a%b] rs' = filter (<1) $ map (\x -> denominator x % numerator x) rs

## 4 Problem 34

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

Solution:

import Data.Char problem_34 = sum [ x | x <- [3..100000], x == facsum x ] where facsum = sum . map (product . enumFromTo 1 . digitToInt) . show

## 5 Problem 35

How many circular primes are there below one million?

Solution:

import Data.List (tails, (\\)) primes :: [Integer] primes = 2 : filter ((==1) . length . primeFactors) [3,5..] primeFactors :: Integer -> [Integer] primeFactors n = factor n primes where factor _ [] = [] factor m (p:ps) | p*p > m = [m] | m `mod` p == 0 = p : factor (m `div` p) (p:ps) | otherwise = factor m ps isPrime :: Integer -> Bool isPrime 1 = False isPrime n = case (primeFactors n) of (_:_:_) -> False _ -> True permutations :: Integer -> [Integer] permutations n = take l $ map (read . take l) $ tails $ take (2*l -1) $ cycle s where s = show n l = length s circular_primes :: [Integer] -> [Integer] circular_primes [] = [] circular_primes (x:xs) | all isPrime p = x : circular_primes xs | otherwise = circular_primes xs where p = permutations x problem_35 :: Int problem_35 = length $ circular_primes $ takeWhile (<1000000) primes

## 6 Problem 36

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

Solution:

import Numeric import Data.Char showBin = flip (showIntAtBase 2 intToDigit) "" isPalindrome x = x == reverse x problem_36 = sum [x | x <- [1,3..1000000], isPalindrome (show x), isPalindrome (showBin x)]

## 7 Problem 37

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

Solution:

import Data.List (tails, inits, nub) primes :: [Integer] primes = 2 : filter ((==1) . length . primeFactors) [3,5..] primeFactors :: Integer -> [Integer] primeFactors n = factor n primes where factor _ [] = [] factor m (p:ps) | p*p > m = [m] | m `mod` p == 0 = p : factor (m `div` p) (p:ps) | otherwise = factor m ps isPrime :: Integer -> Bool isPrime 1 = False isPrime n = case (primeFactors n) of (_:_:_) -> False _ -> True truncs :: Integer -> [Integer] truncs n = nub . map read $ (take l . tail . tails) s ++ (take l . tail . inits) s where l = length s - 1 s = show n problem_37 = sum $ take 11 [x | x <- dropWhile (<=9) primes, all isPrime (truncs x)]

## 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:

problem_38 = maximum $ catMaybes [result | j <- [1..9999], let p2 = show j ++ show (2*j), let p3 = p2 ++ show (3*j), let p4 = p3 ++ show (4*j), let p5 = p4 ++ show (5*j), let result | isPan p2 = Just p2 | isPan p3 = Just p3 | isPan p4 = Just p4 | isPan p5 = Just p5 | otherwise = Nothing] where isPan s = sort s == "123456789"

Other 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 :: Int 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.

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]

## 10 Problem 40

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

Solution:

problem_40 = (d 1)*(d 10)*(d 100)*(d 1000)*(d 10000)*(d 100000)*(d 1000000) where n = concat [show n | n <- [1..]] d j = Data.Char.digitToInt (n !! (j-1))