# Euler problems/121 to 130

(Difference between revisions)

## 1 Problem 121

Investigate the game of chance involving coloured discs.

Solution:

`problem_121 = undefined`

## 2 Problem 122

Finding the most efficient exponentiation method.

Solution using a depth first search, pretty fast :

```import Data.List
import Data.Array.Diff

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```

## 3 Problem 123

Determining the remainder when (pn − 1)n + (pn + 1)n is divided by pn2.

Solution:

```primes = 2 : filter ((==1) . length . primeFactors) [3,5..]

primeFactors n = factor n primes
where
factor _ [] = []
factor m (p:ps) | p*p > m        = [m]
| m `mod` p == 0 = p : [m `div` p]
| otherwise      = factor m ps

isPrime :: Integer -> Bool
isPrime 1 = False
isPrime n = case (primeFactors n) of
(_:_:_)   -> False
_         -> True

problem_123 =
let n=2*(a+1)*primes!!(fromInteger a),
n>10^10
]```

## 4 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]]```

## 5 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)```

## 6 Problem 126

Exploring the number of cubes required to cover every visible face on a cuboid.

Solution:

`problem_126 = undefined`

## 7 Problem 127

Investigating the number of abc-hits below a given limit.

Solution:

```import Data.List
import Data.Map(fromList,(!))
rad x= product[a|(a,_)<-fstfac x]
where
(a,b)=divMod x 1000
maxSingFac x
|not\$null ar=last ar
|otherwise=0
where
ar=[a|(a,b)<-x,b==1]
testPrime x p divx= [swap(x,a,b)|
a<-[p2*a1|a1<-[1..n]],
gcd a x==1,
let b=x-a,
gcd b x==1,
gcd b a==1,
]
where
p2=p^2
(n,m)=divMod x p2
swap (a,b,c)
|b<c=[a,b]
|otherwise=[a,c]
test1 x divx
|otherwise=[]
test x
|(length ff>4)=[]
|(maxSingFac ff>1200)=[]
|otherwise= nub d
where
ff=fstfac x
divx=div x ra
ba=div divx 2
c1=takeWhile (<=ba) primes
d=test1 x divx++[b|p<-c1,b<-testPrime x p divx]

groups=1000
p127 k=[m|a<-[1+k*groups..groups*(k+1)],let t=test a,not\$null t,m<-t]
show1 x=foldl (++) "" \$map ((++" \n").show2) x
show2 [a,b]=show a++"  "++show b
-- write file to change bignum to small num
=if (num>99)
then return()
else do appendFile "p127.log" \$(show1\$p127 num)

fstfac x = [(head a ,length a)|a<-group\$primeFactors x]
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
merge xs@(x:xt) ys@(y:yt) = case compare x y of
LT -> x : (merge xt ys)
EQ -> x : (merge xt yt)
GT -> y : (merge xs yt)

diff  xs@(x:xt) ys@(y:yt) = case compare x y of
LT -> x : (diff xt ys)
EQ -> diff xt yt
GT -> diff xs yt

primes, nonprimes :: [Integer]
primes    = [2,3,5] ++ (diff [7,9..] nonprimes)
nonprimes = foldr1 f . map g \$ tail primes
where f (x:xt) ys = x : (merge xt ys)
g p = [ n*p | n <- [p,p+2..]]
split :: Char -> String -> [String]
split = unfoldr . split'

split' :: Char -> String -> Maybe (String, String)
split' c l
| null l = Nothing
| otherwise = Just (h, drop 1 t)
where (h, t) = span (/=c) l

problem_127=do
let y=sum\$map sToInt \$lines x
print  (y-1-100000)```

## 8 Problem 128

Which tiles in the hexagonal arrangement have prime differences with neighbours?

Solution:

`problem_128 = undefined`

## 9 Problem 129

Investigating minimal repunits that divide by n.

Solution:

`problem_129 = undefined`

## 10 Problem 130

Finding composite values, n, for which n−1 is divisible by the length of the smallest repunits that divide it.

Solution:

```import Data.List
mulMod :: Integral a => a -> a -> a -> a
mulMod a b c= (b * c) `rem` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
where
f x n y
| n == 1 = x `mul` y
| r == 0 = f x2 q y
| otherwise = f x2 q (x `mul` y)
where
(q,r) = quotRem n 2
x2 = sq x
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)

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

primes=2:[a|a<-[2..],isPrime a]
isPrime :: Integer -> Bool
isPrime 1 = False
isPrime n = case (primeFactors n) of
(_:_:_)   -> False
_         -> True
fstfac x = [(head a ,length a)|a<-group\$primeFactors x]
fac [(x,y)]=[x^a|a<-[0..y]]
fac (x:xs)=[a*b|a<-fac [x],b<-fac xs]
factors x=fac\$fstfac x
fun x |(not\$null a)=head a
|otherwise=0
where
a=take 1 [n|n<-sort\$factors (x-1),(powMod x 10 n)==1]
problem_130 =sum\$take 25[a|a<-[1..],
not\$isPrime a,
let b=fun a,
b/=0,
mod (a-1) b==0,
mod a 3 /=0]```