Difference between revisions of "Euler problems/111 to 120"
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | isIncreasing' n p |
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− | import Data.List |
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+ | | n == 0 = True |
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− | digits n |
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+ | | p >= p1 = isIncreasing' (n `div` 10) p1 |
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− | {- change 123 to [3,2,1] |
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⚫ | |||
− | -} |
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− | |n<10=[n] |
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− | |otherwise= y:digits x |
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where |
where |
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− | + | p1 = n `mod` 10 |
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+ | |||
− | isdecr x= |
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+ | isIncreasing :: Int -> Bool |
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− | null$filter (\(x, y)->x-y<0)$zip di k |
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+ | isIncreasing n = isIncreasing' (n `div` 10) (n `mod` 10) |
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+ | |||
+ | isDecreasing' n p |
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+ | | n == 0 = True |
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+ | | p <= p1 = isDecreasing' (n `div` 10) p1 |
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⚫ | |||
where |
where |
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− | + | p1 = n `mod` 10 |
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+ | |||
− | k=0:di |
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+ | isDecreasing :: Int -> Bool |
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− | isincr x= |
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+ | isDecreasing n = isDecreasing' (n `div` 10) (n `mod` 10) |
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− | null$filter (\(x, y)->x-y<0)$zip di k |
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+ | |||
⚫ | |||
+ | isBouncy n = not (isIncreasing n) && not (isDecreasing n) |
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− | di=digits x |
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− | k=tail$di++[0] |
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nnn=1500000 |
nnn=1500000 |
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− | num150 =length [x|x<-[1..nnn], |
+ | num150 =length [x|x<-[1..nnn],isBouncy x] |
+ | p112 n nb |
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− | istwo x|isdecr x||isincr x=1 |
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+ | | fromIntegral nnb / fromIntegral n >= 0.99 = n |
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⚫ | |||
+ | | otherwise = prob112' (n+1) nnb |
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⚫ | |||
⚫ | |||
− | if (div n1 n2==100) |
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+ | nnb = if isBouncy n then nb + 1 else nb |
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− | then do appendFile "file.log" ((show n1) ++" "++ (show n2)++"\n") |
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− | return() |
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− | else problem_112 (n1+1) (n2+istwo (n1+1)) |
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− | main= problem_112 nnn num150 |
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⚫ | |||
</haskell> |
</haskell> |
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it is another way to solution this problem: |
it is another way to solution this problem: |
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<haskell> |
<haskell> |
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+ | binomial x y =div (prodxy (y+1) x) (prodxy 1 (x-y)) |
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− | import List |
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+ | prodxy x y=product[x..y] |
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− | series 2 =replicate 10 1 |
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+ | problem_113=sum[binomial (8+a) a+binomial (9+a) a-10|a<-[1..100]] |
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− | series n=sumkey$map (\(x, y)->map (*y) x)$zip key (series (n-1)) |
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− | key =[replicate (a+1) 1++replicate (9-a) 0|a<-[0..9]] |
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− | sumkey k=[sum [a!!m|a<-k]|m<-[0..9]] |
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− | fun x= sum [(sum$series i)-1|i<-[2..x]]-(x-1)*9-1+(sum$series x) |
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− | problem_113 =fun 101 |
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</haskell> |
</haskell> |
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== [http://projecteuler.net/index.php?section=view&id=114 Problem 114] == |
== [http://projecteuler.net/index.php?section=view&id=114 Problem 114] == |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | -- fun in p115 |
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− | slowfibs n |
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⚫ | |||
− | |n<4=1 |
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− | |otherwise=2*slowfibs (n-1)-slowfibs (n-2)+slowfibs(n-4) |
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− | fibs = 1 : 1: 1: 1: zipWith3 (\a b c->2*a-b+c) c b a |
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− | where |
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− | a=fibs |
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− | b=tail$tail fibs |
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− | c=tail$tail$tail fibs |
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− | fast=[fibs!! a|a<-[1..51]] |
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− | test=[slowfibs a|a<-[1..21]] |
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⚫ | |||
</haskell> |
</haskell> |
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prodxy x y=product[x..y] |
prodxy x y=product[x..y] |
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fun m n=sum[binomial (k+a) (k-a)|a<-[0..div (n+1) (m+1)],let k=1-a*m+n] |
fun m n=sum[binomial (k+a) (k-a)|a<-[0..div (n+1) (m+1)],let k=1-a*m+n] |
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− | p114=fun 3 50 |
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problem_115 = (+1)$length$takeWhile (<10^6) [fun 50 i|i<-[1..]] |
problem_115 = (+1)$length$takeWhile (<10^6) [fun 50 i|i<-[1..]] |
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</haskell> |
</haskell> |
Revision as of 05:45, 14 January 2008
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"
Problem 112
Investigating the density of "bouncy" numbers.
Solution:
isIncreasing' n p
| n == 0 = True
| p >= p1 = isIncreasing' (n `div` 10) p1
| otherwise = False
where
p1 = n `mod` 10
isIncreasing :: Int -> Bool
isIncreasing n = isIncreasing' (n `div` 10) (n `mod` 10)
isDecreasing' n p
| n == 0 = True
| p <= p1 = isDecreasing' (n `div` 10) p1
| otherwise = False
where
p1 = n `mod` 10
isDecreasing :: Int -> Bool
isDecreasing n = isDecreasing' (n `div` 10) (n `mod` 10)
isBouncy n = not (isIncreasing n) && not (isDecreasing n)
nnn=1500000
num150 =length [x|x<-[1..nnn],isBouncy x]
p112 n nb
| fromIntegral nnb / fromIntegral n >= 0.99 = n
| otherwise = prob112' (n+1) nnb
where
nnb = if isBouncy n then nb + 1 else nb
problem_112=p112 (nnn+1) num150
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]]
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
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..]]
Problem 116
Investigating the number of ways of replacing square tiles with one of three coloured tiles.
Solution:
problem_116 = undefined
Problem 117
Investigating the number of ways of tiling a row using different-sized tiles.
Solution:
problem_117 = undefined
Problem 118
Exploring the number of ways in which sets containing prime elements can be made.
Solution:
problem_118 = undefined
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$digits m,
n==a]!!29
Problem 120
Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2.
Solution:
import List
primes :: [Integer]
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
primeFactors :: Integer -> [Integer]
primeFactors n = factor n primes
where
factor _ [] = []
factor m (p:ps) | p*p > m = [m]
| m `mod` p == 0 = p : factor (m `div` p) (p:ps)
| otherwise = factor m ps
isPrime :: Integer -> Bool
isPrime 1 = False
isPrime n = case (primeFactors n) of
(_:_:_) -> False
_ -> True
fun x
|even x=x*(x-2)
|not$null$funb x=head$funb x
|odd e=x*(x-1)
|otherwise=2*x*(e-1)
where
e=div x 2
funb x=take 1 [nn*x|
a<-[1,3..x],
let n=div (x-1) 2,
let p=x*a+n,
isPrime p,
let nn=mod (2*(x*a+n)) x
]
problem_120 = sum [fun a|a<-[3..1000]]