# 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:

```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..]
```

## 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:

```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 =
foldl fcmul 0 . combinationsOf p \$ ds
where
fcmul x y
| np y == 0 = x
| otherwise = x + np y * fc ps (ds \\ y)
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
```

## 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
```

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]
```