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Euler problems/111 to 120

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(Added problem_111)
Line 30: Line 30:
 
Solution:
 
Solution:
 
<haskell>
 
<haskell>
problem_112 = undefined
+
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
  +
 
</haskell>
 
</haskell>
   

Revision as of 03:37, 7 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.

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