Difference between revisions of "Euler problems/31 to 40"
CaleGibbard (talk | contribs) |
|||
Line 6: | Line 6: | ||
This is the naive doubly recursive solution. Speed would be greatly improved by use of [[memoization]], dynamic programming, or the closed form. | This is the naive doubly recursive solution. Speed would be greatly improved by use of [[memoization]], dynamic programming, or the closed form. | ||
<haskell> | <haskell> | ||
− | problem_31 = | + | 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) | |
− | |||
− | |||
− | |||
</haskell> | </haskell> | ||
Line 21: | Line 18: | ||
combinations = foldl (\without p -> | combinations = foldl (\without p -> | ||
let (poor,rich) = splitAt p without | let (poor,rich) = splitAt p without | ||
− | with = poor ++ | + | with = poor ++ zipWith (++) (map (map (p:)) with) |
− | + | rich | |
− | |||
in with | in with | ||
) ([[]] : repeat []) | ) ([[]] : repeat []) | ||
− | problem_31 = | + | problem_31 = length $ combinations coins !! 200 |
− | |||
</haskell> | </haskell> | ||
Line 37: | Line 32: | ||
<haskell> | <haskell> | ||
import Control.Monad | import Control.Monad | ||
+ | |||
combs 0 xs = [([],xs)] | combs 0 xs = [([],xs)] | ||
− | combs n xs = [(y:ys,rest)|y<-xs, (ys,rest)<-combs (n-1) (delete y xs)] | + | combs n xs = [(y:ys,rest) | y <- xs, (ys,rest) <- combs (n-1) (delete y xs)] |
l2n :: (Integral a) => [a] -> a | l2n :: (Integral a) => [a] -> a | ||
Line 46: | Line 42: | ||
explode :: (Integral a) => a -> [a] | explode :: (Integral a) => a -> [a] | ||
− | explode = | + | 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_32 = sum pandigiticals | ||
</haskell> | </haskell> | ||
Line 65: | Line 61: | ||
<haskell> | <haskell> | ||
import Data.Ratio | import Data.Ratio | ||
− | problem_33 = denominator | + | problem_33 = denominator . product $ rs |
{- | {- | ||
xy/yz = x/z | xy/yz = x/z | ||
Line 71: | Line 67: | ||
9xz + yz = 10xy | 9xz + yz = 10xy | ||
-} | -} | ||
− | rs=[(10*x+y)%(10*y+z) | | + | 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] | |
− | |||
− | |||
− | |||
</haskell> | </haskell> | ||
Line 88: | Line 81: | ||
<haskell> | <haskell> | ||
--http://www.research.att.com/~njas/sequences/A014080 | --http://www.research.att.com/~njas/sequences/A014080 | ||
− | problem_34 = sum[145, 40585] | + | problem_34 = sum [145, 40585] |
</haskell> | </haskell> | ||
Line 99: | Line 92: | ||
--http://www.research.att.com/~njas/sequences/A068652 | --http://www.research.att.com/~njas/sequences/A068652 | ||
isPrime x | isPrime x | ||
− | |x==1=False | + | | x==1 = False |
− | |x==2=True | + | | x==2 = True |
− | |x==3=True | + | | x==3 = True |
− | |otherwise=millerRabinPrimality x 2 | + | | otherwise = millerRabinPrimality x 2 |
− | permutations n = | + | |
− | + | 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 [] = [] | ||
circular_primes (x:xs) | circular_primes (x:xs) | ||
| all isPrime p = x : circular_primes xs | | all isPrime p = x : circular_primes xs | ||
| otherwise = circular_primes xs | | otherwise = circular_primes xs | ||
− | + | where p = permutations x | |
− | + | ||
− | x=[1,3,7,9] | + | x = [1,3,7,9] |
− | dmm=(\x y->x*10+y) | + | |
− | + | dmm = foldl (\x y->x*10+y) 0 | |
− | + | ||
− | + | xx n = map dmm (replicateM n x) | |
− | + | ||
− | problem_35 = | + | problem_35 = (+13) . length . circular_primes |
− | + | $ [a | a <- concat [xx 3,xx 4,xx 5,xx 6], isPrime a] | |
</haskell> | </haskell> | ||
Line 131: | Line 127: | ||
<haskell> | <haskell> | ||
--http://www.research.att.com/~njas/sequences/A007632 | --http://www.research.att.com/~njas/sequences/A007632 | ||
− | problem_36= | + | problem_36 = sum [0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, |
− | + | 7447, 9009, 15351, 32223, 39993, 53235, | |
− | + | 53835, 73737, 585585] | |
− | |||
</haskell> | </haskell> | ||
Line 144: | Line 139: | ||
-- isPrime in p35 | -- isPrime in p35 | ||
-- http://www.research.att.com/~njas/sequences/A020994 | -- http://www.research.att.com/~njas/sequences/A020994 | ||
− | problem_37 =sum [23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397] | + | problem_37 = sum [23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397] |
</haskell> | </haskell> | ||
Line 158: | Line 153: | ||
| otherwise = mult n (i+1) (vs ++ [show (n * i)]) | | otherwise = mult n (i+1) (vs ++ [show (n * i)]) | ||
− | problem_38 = | + | problem_38 = maximum . map read . filter ((['1'..'9'] ==) .sort) |
− | + | $ [mult n 1 [] | n <- [2..9999]] | |
− | |||
− | |||
</haskell> | </haskell> | ||
Line 171: | Line 164: | ||
<haskell> | <haskell> | ||
--http://www.research.att.com/~njas/sequences/A046079 | --http://www.research.att.com/~njas/sequences/A046079 | ||
− | problem_39 =let t=3*5*7 in floor(2^floor(log(1000/t)/log | + | problem_39 = let t = 3*5*7 |
+ | in floor(2^floor(log(1000/t)/log 2)*t) | ||
</haskell> | </haskell> | ||
Revision as of 19:35, 19 February 2008
Contents
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:
--http://www.research.att.com/~njas/sequences/A014080
problem_34 = sum [145, 40585]
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]
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]
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]
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.
--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)
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]