Difference between revisions of "Euler problems/111 to 120"

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
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:

```find2km :: Integral a => a -> (a,a)
find2km n = f 0 n
where
f k m
| r == 1 = (k,m)
| otherwise = f (k+1) q
where (q,r) = quotRem m 2

millerRabinPrimality :: Integer -> Integer -> Bool
millerRabinPrimality n a
| a <= 1 || a >= n-1 =
error \$ "millerRabinPrimality: a out of range ("
++ show a ++ " for "++ show n ++ ")"
| n < 2 = False
| even n = False
| b0 == 1 || b0 == n' = True
| otherwise = iter (tail b)
where
n' = n-1
(k,m) = find2km n'
b0 = powMod n a m
b = take (fromIntegral k) \$ iterate (squareMod n) b0
iter [] = False
iter (x:xs)
| x == 1 = False
| x == n' = True
| otherwise = iter xs

pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
where
f x n y
| n == 1 = x `mul` y
| r == 0 = f x2 q y
| otherwise = f x2 q (x `mul` y)
where
(q,r) = quotRem n 2
x2 = sq x

mulMod :: Integral a => a -> a -> a -> a
mulMod a b c = (b * c) `mod` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)
--isPrime x=millerRabinPrimality x 2
isPrime x
|x<100=isPrime' x
|otherwise=foldl   (&& )True [millerRabinPrimality x y|y<-[2,3,7,61]]
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

getprimes ""= [[]]
getprimes s1=
[n:f|
let len=length s1,
a<-[1..len],
let b=take a s1,
isPrime n,
let k=getprimes \$drop a s1,
f<-k,
]
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 ()
problem_118a=do
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:

```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)
|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]]
```