# Difference between revisions of "Euler problems/31 to 40"

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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> | ||

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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 | ||

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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]
```