# Euler problems/111 to 120

### From HaskellWiki

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

## Contents |

## 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 [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"

## 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.

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

## 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