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

Line 88: | Line 88: | ||

Note: inc and dec contain the same data, but it seems clearer to duplicate them. | Note: inc and dec contain the same data, but it seems clearer to duplicate them. | ||

+ | it is another way to solution this problem: | ||

+ | <haskell> | ||

+ | 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 | ||

+ | </haskell> | ||

== [http://projecteuler.net/index.php?section=view&id=114 Problem 114] == | == [http://projecteuler.net/index.php?section=view&id=114 Problem 114] == | ||

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

## Revision as of 07:01, 8 December 2007

## Contents

## 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:

```
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:

```
problem_114 = undefined
```

## Problem 115

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

Solution:

```
problem_115 = undefined
```

## 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:

```
problem_119 = undefined
```

## Problem 120

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

Solution:

```
problem_120 = undefined
```