# Euler problems/111 to 120

## 1 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"```

## 2 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```

## 3 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.

## 4 Problem 114

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

Solution:

`problem_114 = undefined`

## 5 Problem 115

Finding a generalisation for the number of ways to fill a row with separated blocks.

Solution:

`problem_115 = undefined`

## 6 Problem 116

Investigating the number of ways of replacing square tiles with one of three coloured tiles.

Solution:

`problem_116 = undefined`

## 7 Problem 117

Investigating the number of ways of tiling a row using different-sized tiles.

Solution:

`problem_117 = undefined`

## 8 Problem 118

Exploring the number of ways in which sets containing prime elements can be made.

Solution:

`problem_118 = undefined`

## 9 Problem 119

Investigating the numbers which are equal to sum of their digits raised to some power.

Solution:

`problem_119 = undefined`

## 10 Problem 120

Finding the maximum remainder when (a − 1)n + (a + 1)n is divided by a2.

Solution:

`problem_120 = undefined`