Difference between revisions of "Euler problems/111 to 120"
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=111 Problem 111] == |
Search for 10-digit primes containing the maximum number of repeated digits. |
Search for 10-digit primes containing the maximum number of repeated digits. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=112 Problem 112] == |
Investigating the density of "bouncy" numbers. |
Investigating the density of "bouncy" numbers. |
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Line 62: | Line 62: | ||
</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=113 Problem 113] == |
How many numbers below a googol (10100) are not "bouncy"? |
How many numbers below a googol (10100) are not "bouncy"? |
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problem_113=sum[binomial (8+a) a+binomial (9+a) a-10|a<-[1..100]] |
problem_113=sum[binomial (8+a) a+binomial (9+a) a-10|a<-[1..100]] |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [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. |
Investigating the number of ways to fill a row with separated blocks that are at least three units long. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [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. |
Finding a generalisation for the number of ways to fill a row with separated blocks. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [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. |
Investigating the number of ways of replacing square tiles with one of three coloured tiles. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=117 Problem 117] == |
Investigating the number of ways of tiling a row using different-sized tiles. |
Investigating the number of ways of tiling a row using different-sized tiles. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [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. |
Exploring the number of ways in which sets containing prime elements can be made. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | find2km :: Integral a => a -> (a,a) |
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− | find2km n = f 0 n |
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− | where |
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− | f k m |
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− | | r == 1 = (k,m) |
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− | | otherwise = f (k+1) q |
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− | where (q,r) = quotRem m 2 |
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− | |||
− | millerRabinPrimality :: Integer -> Integer -> Bool |
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− | millerRabinPrimality n a |
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− | | a <= 1 || a >= n-1 = |
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− | error $ "millerRabinPrimality: a out of range (" |
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− | ++ show a ++ " for "++ show n ++ ")" |
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− | | n < 2 = False |
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− | | even n = False |
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− | | b0 == 1 || b0 == n' = True |
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− | | otherwise = iter (tail b) |
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− | where |
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− | n' = n-1 |
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− | (k,m) = find2km n' |
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− | b0 = powMod n a m |
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− | b = take (fromIntegral k) $ iterate (squareMod n) b0 |
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− | iter [] = False |
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− | iter (x:xs) |
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− | | x == 1 = False |
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− | | x == n' = True |
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− | | otherwise = iter xs |
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− | |||
− | pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a |
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− | pow' _ _ _ 0 = 1 |
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− | pow' mul sq x' n' = f x' n' 1 |
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− | where |
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− | f x n y |
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− | | n == 1 = x `mul` y |
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− | | r == 0 = f x2 q y |
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− | | otherwise = f x2 q (x `mul` y) |
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− | where |
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− | (q,r) = quotRem n 2 |
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− | x2 = sq x |
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− | |||
− | mulMod :: Integral a => a -> a -> a -> a |
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− | mulMod a b c = (b * c) `mod` a |
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− | squareMod :: Integral a => a -> a -> a |
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− | squareMod a b = (b * b) `rem` a |
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− | powMod :: Integral a => a -> a -> a -> a |
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− | powMod m = pow' (mulMod m) (squareMod m) |
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− | --isPrime x=millerRabinPrimality x 2 |
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isPrime x |
isPrime x |
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|x<100=isPrime' x |
|x<100=isPrime' x |
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− | |otherwise=foldl (&& )True [millerRabinPrimality x y|y<-[2 |
+ | |otherwise=foldl (&& )True [millerRabinPrimality x y|y<-[2,7,61]] |
− | primes :: [Integer] |
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− | primes = 2 : filter ((==1) . length . primeFactors) [3,5..] |
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− | |||
− | primeFactors :: Integer -> [Integer] |
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− | primeFactors n = factor n primes |
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− | where |
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− | factor _ [] = [] |
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− | factor m (p:ps) | p*p > m = [m] |
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− | | m `mod` p == 0 = p : factor (m `div` p) (p:ps) |
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− | | otherwise = factor m ps |
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− | |||
− | isPrime' :: Integer -> Bool |
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− | isPrime' 1 = False |
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− | isPrime' n = case (primeFactors n) of |
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− | (_:_:_) -> False |
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− | _ -> True |
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− | |||
getprimes ""= [[]] |
getprimes ""= [[]] |
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getprimes s1= |
getprimes s1= |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [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. |
Investigating the numbers which are equal to sum of their digits raised to some power. |
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</haskell> |
</haskell> |
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− | == [http://projecteuler.net/index.php?section= |
+ | == [http://projecteuler.net/index.php?section=problems&id=120 Problem 120] == |
Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2. |
Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2. |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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+ | fun m=div (m*(8*m^2-3*m-5)) 3 |
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− | import List |
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⚫ | |||
− | primes :: [Integer] |
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− | primes = 2 : filter ((==1) . length . primeFactors) [3,5..] |
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− | |||
− | primeFactors :: Integer -> [Integer] |
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− | primeFactors n = factor n primes |
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− | where |
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− | factor _ [] = [] |
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− | factor m (p:ps) | p*p > m = [m] |
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− | | m `mod` p == 0 = p : factor (m `div` p) (p:ps) |
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− | | otherwise = factor m ps |
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− | |||
− | isPrime :: Integer -> Bool |
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− | isPrime 1 = False |
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− | isPrime n = case (primeFactors n) of |
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− | (_:_:_) -> False |
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− | _ -> True |
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− | fun x |
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− | |even x=x*(x-2) |
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− | |not$null$funb x=head$funb x |
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− | |odd e=x*(x-1) |
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− | |otherwise=2*x*(e-1) |
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− | where |
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− | e=div x 2 |
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− | |||
− | funb x=take 1 [nn*x| |
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− | a<-[1,3..x], |
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− | let n=div (x-1) 2, |
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− | let p=x*a+n, |
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− | isPrime p, |
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− | let nn=mod (2*(x*a+n)) x |
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− | ] |
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− | |||
⚫ | |||
</haskell> |
</haskell> |
Revision as of 14:43, 26 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:
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
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
Problem 118
Exploring the number of ways in which sets containing prime elements can be made.
Solution:
isPrime x
|x<100=isPrime' x
|otherwise=foldl (&& )True [millerRabinPrimality x y|y<-[2,7,61]]
getprimes ""= [[]]
getprimes s1=
[n:f|
let len=length s1,
a<-[1..len],
let b=take a s1,
let n=read b::Integer,
isPrime n,
let k=getprimes $drop a s1,
f<-k,
a==len|| n<head f
]
perms :: [a] -> [[a]]
perms [] = [ [] ]
perms (x:xs) =
concat (map (between x) (perms xs))
where
between e [] = [ [e] ]
between e (y:ys) = (e:y:ys) : map (y:) (between e ys)
fun x=do
let cs=length$getprimes x
if (cs/=0) then
appendFile "p118.log"$(++"\n")$show cs
else
return ()
sToInt =(+0).read
problem_118a=do
s<-readFile "p118.log"
print$sum$map sToInt$lines s
main=do
mapM_ fun $perms ['1'..'9']
problem_118a
problem_118 = main
Problem 119
Investigating the numbers which are equal to sum of their digits raised to some power.
Solution:
import Data.List
digits n
{- 123->[3,2,1]
-}
|n<10=[n]
|otherwise= y:digits x
where
(x,y)=divMod n 10
problem_119 =sort [(a^b)|
a<-[2..200],
b<-[2..9],
let m=a^b,
let n=sum$digits m,
n==a]!!29
Problem 120
Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2.
Solution:
fun m=div (m*(8*m^2-3*m-5)) 3
problem_120 = fun 500