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Euler problems/111 to 120

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Do them on your own!
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== [http://projecteuler.net/index.php?section=problems&id=111 Problem 111] ==
  +
Search for 10-digit primes containing the maximum number of repeated digits.
  +
  +
Solution:
  +
<haskell>
  +
import Control.Monad (replicateM)
  +
  +
-- All ways of interspersing n copies of x into a list
  +
intr :: Int -> a -> [a] -> [[a]]
  +
intr 0 _ y = [y]
  +
intr n x (y:ys) = concat
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[map ((replicate i x ++) . (y :)) $ intr (n-i) x ys
  +
| i <- [0..n]]
  +
intr n x _ = [replicate n x]
  +
  +
-- All 10-digit primes containing the maximal number of the digit d
  +
maxDigits :: Char -> [Integer]
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maxDigits d = head $ dropWhile null
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[filter isPrime $ map read $ filter ((/='0') . head) $
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concatMap (intr (10-n) d) $
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replicateM n $ delete d "0123456789"
  +
| n <- [1..9]]
  +
  +
problem_111 = sum $ concatMap maxDigits "0123456789"
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=112 Problem 112] ==
  +
Investigating the density of "bouncy" numbers.
  +
  +
Solution:
  +
<haskell>
  +
import Data.List
  +
  +
isIncreasing x = show x == sort (show x)
  +
isDecreasing x = reverse (show x) == sort (show x)
  +
isBouncy x = not (isIncreasing x) && not (isDecreasing x)
  +
  +
findProportion prop = snd . head . filter condition . zip [1..]
  +
where condition (a,b) = a >= prop * fromIntegral b
  +
  +
problem_112 = findProportion 0.99 $ filter isBouncy [1..]
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=113 Problem 113] ==
  +
How many numbers below a googol (10100) are not "bouncy"?
  +
  +
Solution:
  +
<haskell>
  +
import Array
  +
  +
mkArray b f = listArray b $ map f (range b)
  +
  +
digits = 100
  +
  +
inc = mkArray ((1, 0), (digits, 9)) ninc
  +
dec = mkArray ((1, 0), (digits, 9)) ndec
  +
  +
ninc (1, _) = 1
  +
ninc (l, d) = sum [inc ! (l-1, i) | i <- [d..9]]
  +
  +
ndec (1, _) = 1
  +
ndec (l, d) = sum [dec ! (l-1, i) | i <- [0..d]]
  +
  +
problem_113 = sum [inc ! i | i <- range ((digits, 0), (digits, 9))]
  +
+ sum [dec ! i | i <- range ((1, 1), (digits, 9))]
  +
- digits*9 -- numbers like 11111 are counted in both inc and dec
  +
- 1 -- 0 is included in the increasing numbers
  +
</haskell>
  +
Note: inc and dec contain the same data, but it seems clearer to duplicate them.
  +
  +
it is another way to solution this problem:
  +
<haskell>
  +
binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
  +
prodxy x y=product[x..y]
  +
problem_113=sum[binomial (8+a) a+binomial (9+a) a-10|a<-[1..100]]
  +
</haskell>
  +
== [http://projecteuler.net/index.php?section=problems&id=114 Problem 114] ==
  +
Investigating the number of ways to fill a row with separated blocks that are at least three units long.
  +
  +
Solution:
  +
<haskell>
  +
-- fun in p115
  +
problem_114=fun 3 50
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=115 Problem 115] ==
  +
Finding a generalisation for the number of ways to fill a row with separated blocks.
  +
  +
Solution:
  +
<haskell>
  +
binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
  +
prodxy x y=product[x..y]
  +
fun m n=sum[binomial (k+a) (k-a)|a<-[0..div (n+1) (m+1)],let k=1-a*m+n]
  +
problem_115 = (+1)$length$takeWhile (<10^6) [fun 50 i|i<-[1..]]
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=116 Problem 116] ==
  +
Investigating the number of ways of replacing square tiles with one of three coloured tiles.
  +
  +
Solution:
  +
<haskell>
  +
binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
  +
prodxy x y=product[x..y]
  +
f116 n x=sum[binomial (a+b) a|a<-[1..div n x],let b=n-a*x]
  +
p116 x=sum[f116 x a|a<-[2..4]]
  +
problem_116 = p116 50
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=117 Problem 117] ==
  +
Investigating the number of ways of tiling a row using different-sized tiles.
  +
  +
Solution:
  +
<haskell>
  +
fibs5 = 0 : 0 :1: 1:zipWith4 (\a b c d->a+b+c+d) fibs5 a1 a2 a3
  +
where
  +
a1=tail fibs5
  +
a2=tail a1
  +
a3=tail a2
  +
p117 x=fibs5!!(x+2)
  +
problem_117 = p117 50
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=118 Problem 118] ==
  +
Exploring the number of ways in which sets containing prime elements can be made.
  +
  +
Solution:
  +
<haskell>
  +
digits = ['1'..'9']
  +
  +
-- possible partitions voor prime number sets
  +
-- leave out patitions with more than 4 1's
  +
-- because only {2,3,5,7,..} is possible
  +
-- and the [9]-partition because every permutation of all
  +
-- nine digits is divisable by 3
  +
test xs
  +
|len>4=False
  +
|xs==[9]=False
  +
|otherwise=True
  +
where
  +
len=length $filter (==1) xs
  +
parts = filter test $partitions 9
  +
permutationsOf [] = [[]]
  +
permutationsOf xs = [x:xs' | x <- xs, xs' <- permutationsOf (delete x xs)]
  +
combinationsOf 0 _ = [[]]
  +
combinationsOf _ [] = []
  +
combinationsOf k (x:xs) =
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map (x:) (combinationsOf (k-1) xs) ++ combinationsOf k xs
  +
  +
priemPerms [] = 0
  +
priemPerms ds =
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fromIntegral . length . filter (isPrime . read) . permutationsOf $ ds
  +
setsums [] 0 = [[]]
  +
setsums [] _ = []
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setsums (x:xs) n
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| x > n = setsums xs n
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| otherwise = map (x:) (setsums (x:xs) (n-x)) ++ setsums xs n
  +
  +
partitions n = setsums (reverse [1..n]) n
  +
  +
fc :: [Integer] -> [Char] -> Integer
  +
fc (p:[]) ds = priemPerms ds
  +
fc (p:ps) ds =
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sum [np y * fc ps (ds \\ y) | y <- combinationsOf p ds, np y /= 0]
  +
where
  +
np = priemPerms
  +
-- here is the 'imperfection' correction method:
  +
-- make use of duplicate reducing factors for partitions
  +
-- with repeating factors, f.i. [1,1,1,1,2,3]:
  +
-- in this case 4 1's -> factor = 4!
  +
-- or for [1,1,1,3,3] : factor = 3! * 2!
  +
dupF :: [Integer] -> Integer
  +
dupF = product . map (product . enumFromTo 1 . fromIntegral . length) . group
  +
  +
main = do
  +
print . sum . map (\x -> fc x digits `div` dupF x) $ parts
  +
problem_118 = main
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=119 Problem 119] ==
  +
Investigating the numbers which are equal to sum of their digits raised to some power.
  +
  +
Solution:
  +
<haskell>
  +
import Data.List
  +
digits n
  +
{- 123->[3,2,1]
  +
-}
  +
|n<10=[n]
  +
|otherwise= y:digits x
  +
where
  +
(x,y)=divMod n 10
  +
problem_119 =sort [(a^b)|
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a<-[2..200],
  +
b<-[2..9],
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let m=a^b,
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let n=sum$digits m,
  +
n==a]!!29
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=120 Problem 120] ==
  +
Finding the maximum remainder when (a − 1)<sup>n</sup> + (a + 1)<sup>n</sup> is divided by a<sup>2</sup>.
  +
  +
Solution:
  +
<haskell>
  +
fun m=div (m*(8*m^2-3*m-5)) 3
  +
problem_120 = fun 500
  +
</haskell>
  +
  +
  +
I have no idea what the above solution has to do with this
  +
problem, even though it produces the correct answer. I suspect
  +
it is some kind of red herring. Below you will find a more holy
  +
mackerel approach, based on the observation that:
  +
  +
1. (a-1)<sup>n</sup> + (a+1)<sup>n</sup> = 2 if n is odd, and 2an if n is even (mod a<sup>2</sup>)
  +
  +
2. the maximum of 2an mod a<sup>2</sup> occurs when n = (a-1)/2
  +
  +
I hope this is a little more transparent than the solution
  +
proposed above. Henrylaxen Mar 5, 2008
  +
  +
<haskell>
  +
maxRemainder n = 2 * n * ((n-1) `div` 2)
  +
problem_120 = sum $ map maxRemainder [3..1000]
  +
</haskell>

Latest revision as of 08:07, 23 February 2010

Contents

[edit] 1 Problem 111

Search for 10-digit primes containing the maximum number of repeated digits.

Solution:

import Control.Monad (replicateM)
 
-- All ways of interspersing n copies of x into a list
intr :: Int -> a -> [a] -> [[a]]
intr 0 _ y      = [y]
intr n x (y:ys) = concat
                  [map ((replicate i x ++) . (y :)) $ intr (n-i) x ys
                       | i <- [0..n]]
intr n x _      = [replicate n x]
 
-- All 10-digit primes containing the maximal number of the digit d
maxDigits :: Char -> [Integer]
maxDigits d = head $ dropWhile null
              [filter isPrime $ map read $ filter ((/='0') . head) $
               concatMap (intr (10-n) d) $
               replicateM n $ delete d "0123456789"
                   | n <- [1..9]]
 
problem_111 = sum $ concatMap maxDigits "0123456789"

[edit] 2 Problem 112

Investigating the density of "bouncy" numbers.

Solution:

import Data.List
 
isIncreasing x = show x == sort (show x)
isDecreasing x = reverse (show x) == sort (show x)
isBouncy x = not (isIncreasing x) && not (isDecreasing x)
 
findProportion prop = snd . head . filter condition . zip [1..]
  where condition (a,b) = a >= prop * fromIntegral b
 
problem_112 = findProportion 0.99 $ filter isBouncy [1..]

[edit] 3 Problem 113

How many numbers below a googol (10100) are not "bouncy"?

Solution:

import Array
 
mkArray b f = listArray b $ map f (range b)
 
digits = 100
 
inc = mkArray ((1, 0), (digits, 9)) ninc
dec = mkArray ((1, 0), (digits, 9)) ndec
 
ninc (1, _) = 1
ninc (l, d) = sum [inc ! (l-1, i) | i <- [d..9]]
 
ndec (1, _) = 1
ndec (l, d) = sum [dec ! (l-1, i) | i <- [0..d]]
 
problem_113 = sum [inc ! i | i <- range ((digits, 0), (digits, 9))]
               + sum [dec ! i | i <- range ((1, 1), (digits, 9))]
               - digits*9 -- numbers like 11111 are counted in both inc and dec 
               - 1 -- 0 is included in the increasing numbers

Note: inc and dec contain the same data, but it seems clearer to duplicate them.

it is another way to solution this problem:

binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
prodxy x y=product[x..y]
problem_113=sum[binomial (8+a) a+binomial (9+a) a-10|a<-[1..100]]

[edit] 4 Problem 114

Investigating the number of ways to fill a row with separated blocks that are at least three units long.

Solution:

-- fun in p115
problem_114=fun 3 50

[edit] 5 Problem 115

Finding a generalisation for the number of ways to fill a row with separated blocks.

Solution:

binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
prodxy x y=product[x..y]
fun m n=sum[binomial (k+a) (k-a)|a<-[0..div (n+1) (m+1)],let k=1-a*m+n]
problem_115 = (+1)$length$takeWhile (<10^6) [fun 50 i|i<-[1..]]

[edit] 6 Problem 116

Investigating the number of ways of replacing square tiles with one of three coloured tiles.

Solution:

binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y))
prodxy x y=product[x..y]
f116 n x=sum[binomial (a+b) a|a<-[1..div n x],let b=n-a*x]
p116 x=sum[f116 x a|a<-[2..4]]
problem_116 = p116 50

[edit] 7 Problem 117

Investigating the number of ways of tiling a row using different-sized tiles.

Solution:

fibs5 = 0 : 0 :1: 1:zipWith4 (\a b c d->a+b+c+d) fibs5 a1 a2 a3 
    where
    a1=tail fibs5
    a2=tail a1
    a3=tail a2
p117 x=fibs5!!(x+2)
problem_117 = p117 50

[edit] 8 Problem 118

Exploring the number of ways in which sets containing prime elements can be made.

Solution:

digits = ['1'..'9']
 
-- possible partitions voor prime number sets
-- leave out patitions with more than 4 1's 
-- because only {2,3,5,7,..} is possible
-- and the [9]-partition because every permutation of all
-- nine digits is divisable by 3
test xs
    |len>4=False
    |xs==[9]=False
    |otherwise=True
    where
    len=length $filter (==1) xs
parts = filter test $partitions  9
permutationsOf [] = [[]]
permutationsOf xs = [x:xs' | x <- xs, xs' <- permutationsOf (delete x xs)]
combinationsOf  0 _ = [[]]
combinationsOf  _ [] = []
combinationsOf  k (x:xs) =
    map (x:) (combinationsOf (k-1) xs) ++ combinationsOf k xs
 
priemPerms [] = 0
priemPerms ds = 
    fromIntegral . length . filter (isPrime . read) . permutationsOf $ ds
setsums [] 0 = [[]]
setsums [] _ = []
setsums (x:xs) n 
    | x > n     = setsums xs n
    | otherwise = map (x:) (setsums (x:xs) (n-x)) ++ setsums xs n
 
partitions n = setsums (reverse [1..n]) n
 
fc :: [Integer] -> [Char] -> Integer
fc (p:[]) ds = priemPerms ds
fc (p:ps) ds = 
    sum [np y * fc ps (ds \\ y) | y <- combinationsOf p ds, np y /= 0]
        where
        np = priemPerms 
-- here is the 'imperfection' correction method:
-- make use of duplicate reducing factors for partitions
-- with repeating factors, f.i. [1,1,1,1,2,3]: 
-- in this case 4 1's -> factor = 4!
-- or for [1,1,1,3,3] : factor = 3! * 2!
dupF :: [Integer] -> Integer
dupF = product . map (product . enumFromTo 1 . fromIntegral . length) . group
 
main = do
    print . sum . map (\x -> fc x digits `div` dupF x) $ parts 
problem_118 = main

[edit] 9 Problem 119

Investigating the numbers which are equal to sum of their digits raised to some power.

Solution:

import Data.List
digits n 
{-  123->[3,2,1]
 -}
    |n<10=[n]
    |otherwise= y:digits x 
    where
    (x,y)=divMod n 10
problem_119 =sort [(a^b)|
    a<-[2..200],
    b<-[2..9],
    let m=a^b,
    let n=sum$digits m,
    n==a]!!29

[edit] 10 Problem 120

Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2.

Solution:

fun m=div (m*(8*m^2-3*m-5)) 3
problem_120 = fun 500


I have no idea what the above solution has to do with this problem, even though it produces the correct answer. I suspect it is some kind of red herring. Below you will find a more holy mackerel approach, based on the observation that:

1. (a-1)n + (a+1)n = 2 if n is odd, and 2an if n is even (mod a2)

2. the maximum of 2an mod a2 occurs when n = (a-1)/2

I hope this is a little more transparent than the solution proposed above. Henrylaxen Mar 5, 2008

maxRemainder n = 2 * n * ((n-1) `div` 2)
problem_120 = sum $ map maxRemainder [3..1000]