Euler problems/111 to 120

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Revision as of 18:54, 12 November 2007 by YitzGale (talk | contribs) (Added problem_111)

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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:

problem_112 = undefined

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.

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