# Euler problems/121 to 130

(Difference between revisions)

## 1 Problem 121

Investigate the game of chance involving coloured discs.

Solution:

```import Data.List
problem_121 = possibleGames `div` winningGames
where
possibleGames = product [1..16]
winningGames =
(1+) \$ sum \$ map product \$ chooseUpTo 7 [1..15]
chooseUpTo 0     _ = []
chooseUpTo (n+1) x =
[y:z |
(y:ys) <- tails x,
z <- []: chooseUpTo n ys
]```

## 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:

```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
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.Array.IArray
import Data.Array.Unboxed

main = appendFile "p127.log" \$show\$ solve 99999

rad x = fromIntegral \$ product \$ map fst \$ primePowerFactors \$ fromIntegral x
primePowerFactors x = [(head a ,length a)|a<-group\$primeFactors x]
solve :: Int -> Int
solve n = sum [ c | (rc,c) <- invrads
, 2 * rc < c
, (ra, a) <- takeWhile (\(a,_)->(c > 2*rc*a)) invrads
, a < c `div` 2
, gcd ra rc == 1
, ra * rads ! (c - a) < c `div` rc]
where
invrads = sort \$ map (\(a,b) -> (b, a)) \$ assocs rads
problem_127 = main```

## 8 Problem 128

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

Solution:

```p128=
concat[m|a<-[0..70000],let m=middle a++right a,not\$null m]
where
middle n
|all isPrime [11+6*n,13+6*n,29+12*n]=[2+3*(n+1)*(n+2)]
|otherwise=[]
right n
|all isPrime [11+6*n,17+6*n,17+12*n]=[1+3*(n+2)*(n+3)]
|otherwise=[]
problem_128=do
print(p128!!1997)
isPrime x
|x<100=isPrime' x
|otherwise=all (millerRabinPrimality x )[2,7,61]```

## 9 Problem 129

Investigating minimal repunits that divide by n.

Solution:

```import Data.List
factors x=fac\$fstfac x
funp (p,1)=head[a|a<-sort\$factors (p-1),powMod p 10 a==1]
funp (p,s)=p^(s-1)*funp (p,1)
funn []=1
funn (x:xs) =lcm (funp x) (funn xs)
p129 q=head[a|a<-[q..],gcd a 10==1,let s=funn\$fstfac\$(*9) a,s>q,s>a]
problem_129 = p129 (10^6)```

## 10 Problem 130

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

Solution:

```--factors powMod in p129