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

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## 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
```