Euler problems/121 to 130
Problem 121
Investigate the game of chance involving coloured discs.
Solution:
problem_121 = undefined
Problem 122
Finding the most efficient exponentiation method.
Solution using a depth first search, pretty fast :
import Data.List
import Data.Array.Diff
import Control.Monad
depthAddChain 12 branch mins = mins
depthAddChain d branch mins = foldl' step mins $ nub $ filter (> head branch)
$ liftM2 (+) branch branch
where
step da e | e > 200 = da
| otherwise =
case compare (da ! e) d of
GT -> depthAddChain (d+1) (e:branch) $ da // [(e,d)]
EQ -> depthAddChain (d+1) (e:branch) da
LT -> da
baseBranch = [2,1]
baseMins :: DiffUArray Int Int
baseMins = listArray (1,200) $ 0:1: repeat maxBound
problem_122 = sum . elems $ depthAddChain 2 baseBranch baseMins
Problem 123
Determining the remainder when (pn − 1)n + (pn + 1)n is divided by pn2.
Solution:
problem_123 = undefined
Problem 124
Determining the kth element of the sorted radical function.
Solution:
import List
primes :: [Integer]
primes = 2 : filter ((==1) . length . primeFactors) [3,5..]
primeFactors :: Integer -> [Integer]
primeFactors n = factor n primes
where
factor _ [] = []
factor m (p:ps) | p*p > m = [m]
| m `mod` p == 0 = p : factor (m `div` p) (p:ps)
| otherwise = factor m ps
problem_124=snd$(!!9999)$sort[(product$nub$primeFactors x,x)|x<-[1..100000]]
Problem 125
Finding square sums that are palindromic.
Solution:
import Data.List
import Data.Map(fromList,(!))
toFloat = (flip encodeFloat 0)
digits n
{- 123->[3,2,1]
-}
|n<10=[n]
|otherwise= y:digits x
where
(x,y)=divMod n 10
palind n=foldl dmm 0 (digits n)
-- 123 ->321
dmm=(\x y->x*10+y)
makepalind n=(n*d+p):[c+b*d|b<-[0..9]]
where
a=(+1)$floor$logBase 10$fromInteger n
d=10^a
p=palind n
c=n*10*d+p
twomakep n=(n*d+p)
where
a=(+1)$floor$logBase 10$fromInteger n
d=10^a
p=palind n
p125=sum[b|a<-[1..999], b<-makepalind a,not$null$ funa b]
p125a=sum[b|a<-[1000..9999], let b=twomakep a,not$null$ funa b]
p125b=sum[a|a<-[1..9], not$null$ funa a]
findmap=fromList[(a,2*fill_map a)|a<-[0..737]]
fill_map x
|odd x=fastsum $div (x-1) 2
|otherwise=fastsumodd (x-1)
where
fastsum y=div (y*(y+1)*(2*y+1)) 6
fastsumodd y=let n=div (y+1) 2 in div (n*(4*n*n-1)) 3
funa x=[(a,x)|a<-takeWhile (\a->a*a*a<4*x) [2..],funb a x]
funb x n
|odd x=d2==0 && 4*d1>=(x+1)^2 && isSq d1
|otherwise=d4==0 && odd d3 && d3>=(x+1)^2 && isSq d3
where
x1=fromInteger x
(d1,d2)=divMod ((n-findmap! x1)) (x)
(d3,d4)=divMod ((4*n-findmap!x1)) (x)
isSq x=(floor$sqrt$toFloat x)^2==x
problem_125 = (p125+p125a+p125b)
Problem 126
Exploring the number of cubes required to cover every visible face on a cuboid.
Solution:
problem_126 = undefined
Problem 127
Investigating the number of abc-hits below a given limit.
Solution:
problem_127 = undefined
Problem 128
Which tiles in the hexagonal arrangement have prime differences with neighbours?
Solution:
problem_128 = undefined
Problem 129
Investigating minimal repunits that divide by n.
Solution:
problem_129 = undefined
Problem 130
Finding composite values, n, for which n−1 is divisible by the length of the smallest repunits that divide it.
Solution:
problem_130 = undefined