Add solutions for problems 28-32.

Euler.hs

@@ -12,17 +12,19 @@ module Euler
   , primes
   , zipArraysWith
   , RangeIx(..)
-  , digitsOf
   , divisors
   , properDivisors
   , toDecimal
   , fromDecimal
+  , toDigits
+  , fromDigits
   , module Control.Applicative
   , module Control.Arrow
   , module Control.Monad
   , module Data.Function
   , module Data.List
   , module Data.Ratio
+  , module Data.Tuple
   , module Data.Word
   ) where
 
@@ -36,6 +38,7 @@ import Data.Array.Unboxed
 import Data.Function
 import Data.List
 import Data.Ratio
+import Data.Tuple
 import Data.Word
 
 import qualified Control.Monad.ST.Lazy as LST
@@ -77,9 +80,6 @@ zipArraysWith :: (IArray arrA a, IArray arrB b, IArray arrC c, RangeIx i)
 zipArraysWith f as bs = array newRange $ [ (i, f (as!i) (bs!i)) | i <- range newRange ]
    where newRange = intersectBounds (bounds as) (bounds bs)
 
-digitsOf :: (Read a, Show a, Integral a) => a -> [a]
-digitsOf = map (read . (:[])) . show
-
 divisors :: Integral a => a -> [a]
 divisors n = nub $ concat [ [m, q] | m <- takeWhile (\x -> x^2 <= n) [1..], let (q, r) = n `divMod` m, r == 0 ]
 
@@ -105,7 +105,6 @@ toDecimal' n m = first (map fst) $ second (map fst) $ collect [] xs
 fromDecimal :: Decimal -> Rational
 fromDecimal (Decimal n ps rs) = (n % 1) + (p + r) * (1 % 10 ^ length ps)
    where (p, r) = (fromDigits ps % 1, fromDigits rs % (10 ^ length rs - 1))
-         fromDigits = foldl' (\s d -> 10 * s + fromIntegral d) 0
 
 deriving instance Eq Decimal
 deriving instance Ord Decimal
@@ -116,3 +115,10 @@ instance Show Decimal where
       | null rs         = show n ++ "." ++ concat (map show ps)
       | otherwise       = show n ++ "." ++ concat (map show ps)
                                  ++ "(" ++ concat (map show rs) ++ ")"
+
+toDigits :: (Integral a, Integral b) => a -> [b]
+toDigits = reverse . unfoldr (\x ->
+   if x == 0 then Nothing else Just (first fromIntegral $ swap (x `divMod` 10)))
+
+fromDigits :: (Integral a, Integral b) => [a] -> b
+fromDigits = foldl' (\a b -> 10 * a + fromIntegral b) 0

Problem020.hs

@@ -1,2 +1,2 @@
 import Euler
-main = print $ sum $ digitsOf $ product [1..100]
+main = print $ sum $ toDigits $ product [1..100]

Problem028.hs

@@ -0,0 +1,23 @@
+-- Starting with the number 1 and moving to the right in a clockwise direction
+-- a 5 by 5 spiral is formed as follows:
+--
+--   21 22 23 24 25
+--   20  7  8  9 10
+--   19  6  1  2 11
+--   18  5  4  3 12
+--   17 16 15 14 13
+--
+-- It can be verified that the sum of the numbers on the diagonals is 101.
+--
+-- What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral
+-- formed in the same way?
+
+ne = 1 : zipWith (+) nw [2,4..]
+se =     zipWith (+) ne [2,4..]
+sw =     zipWith (+) se [2,4..]
+nw =     zipWith (+) sw [2,4..]
+
+-- Sum of diagonals of an m by m spiral; m = 2*n + 1.
+diagSum n = sum (take (n+1) ne ++ take n se ++ take n sw ++ take n nw)
+
+main = print $ (diagSum 2, diagSum 500)

Problem029.hs

@@ -0,0 +1,4 @@
+-- How many distinct terms are in the sequence generated by a^b
+-- for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?
+import Euler
+main = print $ length $ nub [ a^b | a <- [2..100], b <- [2..100] ]

Problem030.hs

@@ -0,0 +1,4 @@
+-- Find the sum of all the numbers that can be written as the sum of fifth
+-- powers of their digits.
+import Euler
+main = print $ sum $ filter (\n -> sum (map (^5) (toDigits n)) == n) [2..(6 * 9^5)]

Problem031.hs

@@ -0,0 +1,19 @@
+-- In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation:
+--
+--   1p, 2p, 5p, 10p, 20p, 50p, £1 (100p) and £2 (200p)
+--
+-- How many different ways can £2 be made using any number of coins?
+
+import Data.Array
+import Euler
+
+coins = listArray (0,7) [1,2,5,10,20,50,100,200] :: Array Int Int
+
+nWays total = arr ! (total, snd (bounds coins))
+   where
+      arr = array ((0, fst (bounds coins)), (total, snd (bounds coins)))
+         [ ((n, c), sum [ arr ! (n - (coins!c'), c') + if n == coins!c' then 1 else 0
+                        | c' <- [fst (bounds coins)..c], coins!c' <= n ])
+         | n <- [0..total], c <- indices coins ]
+
+main = print $ nWays 200

Problem032.hs

@@ -0,0 +1,35 @@
+-- We shall say that an n-digit number is pandigital if it makes use of all the
+-- digits 1 to n exactly once; for example, the 5-digit number, 15234, is 1
+-- through 5 pandigital.
+--
+-- The product 7254 is unusual, as the identity, 39 × 186 = 7254, containing
+-- multiplicand, multiplier, and product is 1 through 9 pandigital.
+--
+-- Find the sum of all products whose multiplicand/multiplier/product identity
+-- can be written as a 1 through 9 pandigital.
+--
+-- HINT: Some products can be obtained in more than one way so be sure to only
+-- include it once in your sum.
+
+import Euler
+
+partitions []     = [([],[])]
+partitions (x:xs) = map (first (x:)) ps ++ map (second (x:)) ps
+   where ps = partitions xs
+
+main = print $ sum $ nub $ do
+   (as, as') <- partitions [1..9]
+   guard $ not $ null as
+   (bs, cs)  <- partitions as'
+   guard $ not $ null bs
+   guard $ not $ null cs
+   guard $ length as <= length bs
+   guard $ length as <= length cs
+   guard $ length bs <= length cs
+   guard $ length as + length bs >= length cs
+   x <- fromDigits <$> permutations as
+   y <- fromDigits <$> permutations bs
+   guard $ x < y
+   let p = x * y
+   guard $ sort (toDigits p) == cs
+   return p