Euler problems/111 to 120
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(Removing category tags. See Talk:Euler_problems) |
(Added problem_111) |
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Solution: | Solution: | ||
<haskell> | <haskell> | ||
| - | problem_111 = | + | 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" | ||
</haskell> | </haskell> | ||
Revision as of 18:54, 12 November 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:
problem_112 = undefined
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
