# 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
digits n
{-  change 123 to [3,2,1]
-}
|n<10=[n]
|otherwise= y:digits x
where
(x,y)=divMod n 10
isdecr x=
null\$filter (\(x, y)->x-y<0)\$zip di k
where
di=digits x
k=0:di
isincr x=
null\$filter (\(x, y)->x-y<0)\$zip di k
where
di=digits x
k=tail\$di++[0]
nnn=1500000
num150 =length [x|x<-[1..nnn],isdecr x||isincr x]
istwo x|isdecr x||isincr x=1
|otherwise=0
problem_112 n1 n2=
if (div n1 n2==100)
then do appendFile "file.log" ((show n1)  ++"   "++ (show n2)++"\n")
return()
else  problem_112 (n1+1) (n2+istwo (n1+1))
main=  problem_112 nnn 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:

```import List
series  2 =replicate 10 1
series n=sumkey\$map (\(x, y)->map (*y) x)\$zip key (series (n-1))
key =[replicate (a+1) 1++replicate (9-a) 0|a<-[0..9]]
sumkey k=[sum [a!!m|a<-k]|m<-[0..9]]
fun x= sum [(sum\$series i)-1|i<-[2..x]]-(x-1)*9-1+(sum\$series  x)
problem_113 =fun 101
```

## Problem 114

Investigating the number of ways to fill a row with separated blocks that are at least three units long.

Solution:

```slowfibs n
|n<4=1
|otherwise=2*slowfibs (n-1)-slowfibs (n-2)+slowfibs(n-4)
fibs = 1 : 1: 1: 1: zipWith3 (\a b c->2*a-b+c) c b a
where
a=fibs
b=tail\$tail fibs
c=tail\$tail\$tail fibs
fast=[fibs!! a|a<-[1..51]]
test=[slowfibs a|a<-[1..21]]
problem_114=fibs!!51
```

## 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]
p114=fun 3 50
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)
|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]]
```