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Euler problems/151 to 160

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Do them on your own!
+
== [http://projecteuler.net/index.php?section=problems&id=151 Problem 151] ==
  +
Paper sheets of standard sizes: an expected-value problem.
  +
  +
Solution:
  +
<haskell>
  +
problem_151 = fun (1,1,1,1)
  +
  +
fun (0,0,0,1) = 0
  +
fun (0,0,1,0) = fun (0,0,0,1) + 1
  +
fun (0,1,0,0) = fun (0,0,1,1) + 1
  +
fun (1,0,0,0) = fun (0,1,1,1) + 1
  +
fun (a,b,c,d) =
  +
(pickA + pickB + pickC + pickD) / (a + b + c + d)
  +
where
  +
pickA | a > 0 = a * fun (a-1,b+1,c+1,d+1)
  +
| otherwise = 0
  +
pickB | b > 0 = b * fun (a,b-1,c+1,d+1)
  +
| otherwise = 0
  +
pickC | c > 0 = c * fun (a,b,c-1,d+1)
  +
| otherwise = 0
  +
pickD | d > 0 = d * fun (a,b,c,d-1)
  +
| otherwise = 0
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=152 Problem 152] ==
  +
Writing 1/2 as a sum of inverse squares
  +
  +
Note that if p is an odd prime, the sum of inverse squares of
  +
all terms divisible by p must have reduced denominator not divisible
  +
by p.
  +
  +
Solution:
  +
<haskell>
  +
import Data.Ratio
  +
import Data.List
  +
import Data.Ord (comparing)
  +
import Data.Function (on)
  +
  +
invSq n = 1 % (n * n)
  +
sumInvSq = sum . map invSq
  +
  +
subsets (x:xs) = let s = subsets xs in s ++ map (x :) s
  +
subsets _ = [[]]
  +
  +
primes = 2 : 3 : 7 : [p | p <- [11, 13..79],
  +
all (\q -> p `mod` q /= 0) [3, 5, 7]]
  +
  +
-- All subsets whose sum of inverse squares,
  +
-- when added to x, does not contain a factor of p
  +
pfree s x p = [(y, t) | t <- subsets s, let y = x + sumInvSq t,
  +
denominator y `mod` p /= 0]
  +
  +
  +
-- All pairs (x, s) where x is a rational number whose reduced
  +
-- denominator is not divisible by any prime greater than 3;
  +
-- and s is all sets of numbers up to 80 divisible
  +
-- by a prime greater than 3, whose sum of inverse squares is x.
  +
only23 = foldl fun [(0, [[]])] [13, 7, 5]
  +
where
  +
fun a p =
  +
collect $ [(y, u ++ v) |
  +
(x, s) <- a,
  +
(y, v) <- pfree (terms p) x p,
  +
u <- s]
  +
terms p =
  +
[n * p |
  +
n <- [1..80`div`p],
  +
all (\q -> n `mod` q /= 0) $
  +
11 : takeWhile (>= p) [13, 7, 5]
  +
]
  +
collect =
  +
map (\z -> (fst $ head z, map snd z)) .
  +
groupBy fstEq . sortBy cmpFst
  +
fstEq = (==) `on` fst
  +
cmpFst = comparing fst
  +
  +
-- All subsets (of an ordered set) whose sum of inverse squares is x
  +
findInvSq x y =
  +
fun x $ zip3 y (map invSq y) (map sumInvSq $ init $ tails y)
  +
where
  +
fun 0 _ = [[]]
  +
fun x ((n, r, s):ns)
  +
| r > x = fun x ns
  +
| s < x = []
  +
| otherwise = map (n :) (fun (x - r) ns) ++ fun x ns
  +
fun _ _ = []
  +
  +
-- All numbers up to 80 that are divisible only by the primes
  +
-- 2 and 3 and are not divisible by 32 or 27.
  +
all23 = [n | a <- [0..4], b <- [0..2], let n = 2^a * 3^b, n <= 80]
  +
  +
solutions =
  +
[sort $ u ++ v |
  +
(x, s) <- only23,
  +
u <- findInvSq (1%2 - x) all23,
  +
v <- s
  +
]
  +
  +
problem_152 = length solutions
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=153 Problem 153] ==
  +
Investigating Gaussian Integers
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=154 Problem 154] ==
  +
Exploring Pascal's pyramid.
  +
  +
{{sect-stub}}
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=155 Problem 155] ==
  +
Counting Capacitor Circuits.
  +
  +
Solution:
  +
<haskell>
  +
--http://www.research.att.com/~njas/sequences/A051389
  +
a051389=
  +
[1, 2, 4, 8, 20, 42,
  +
102, 250, 610, 1486,
  +
3710, 9228, 23050, 57718,
  +
145288, 365820, 922194, 2327914
  +
]
  +
problem_155 = sum a051389
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=156 Problem 156] ==
  +
Counting Digits
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=157 Problem 157] ==
  +
Solving the diophantine equation 1/a+1/b= p/10n
  +
  +
Solution:
  +
<haskell>
  +
-- Call (a,b,p) a primitive tuple of equation 1/a+1/b=p/10^n
  +
-- a and b are divisors of 10^n, gcd a b == 1, a <= b and a*b <= 10^n
  +
-- I noticed that the number of variants with a primitive tuple
  +
-- is equal to the number of divisors of p.
  +
-- So I produced all possible primitive tuples per 10^n and
  +
-- summed all the number of divisors of every p
  +
  +
import Data.List
  +
k `divides` n = n `mod` k == 0
  +
  +
divisors n
  +
| n == 10 = [1,2,5,10]
  +
| otherwise =
  +
[ d |
  +
d <- [1..n `div` 5],
  +
d `divides` n ]
  +
++ [n `div` 4, n `div` 2,n]
  +
fp n =
  +
[ n*(a+b) `div` ab |
  +
a <- ds,
  +
b <- dropWhile (<a) ds,
  +
gcd a b == 1,
  +
let ab = a*b,
  +
ab <= n
  +
]
  +
where
  +
ds = divisors n
  +
numDivisors :: Integer -> Integer
  +
numDivisors n = product [ toInteger (a+1) | (p,a) <- primePowerFactors n]
  +
numVgln = sum . map numDivisors . fp
  +
  +
main = do
  +
print . sum . map numVgln . takeWhile (<=10^9) . iterate (10*) $ 10
  +
primePowerFactors x = [(head a ,length a)|a<-group$primeFactors x]
  +
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..]]
  +
primeFactors n =
  +
factor n primes
  +
where
  +
factor n (p:ps)
  +
| p*p > n = [n]
  +
| n `mod` p == 0 = p : factor (n `div` p) (p:ps)
  +
| otherwise = factor n ps
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=158 Problem 158] ==
  +
Exploring strings for which only one character comes lexicographically after its neighbour to the left.
  +
  +
Solution:
  +
<haskell>
  +
factorial n = product [1..toInteger n]
  +
fallingFactorial x n = product [x - i | i <- [0..fromIntegral n - 1] ]
  +
choose n k = fallingFactorial n k `div` factorial k
  +
fun n=(2 ^ n - n - 1) * choose 26 n
  +
problem_158=maximum$map fun [1..26]
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=159 Problem 159] ==
  +
Digital root sums of factorisations.
  +
  +
Solution:
  +
<haskell>
  +
import Control.Monad
  +
import Data.Array.ST
  +
import qualified Data.Array.Unboxed as U
  +
spfArray :: U.UArray Int Int
  +
spfArray = runSTUArray (do
  +
arr <- newArray (2,m-1) 0
  +
forM_ [2 .. m-1] $ \n ->
  +
writeArray arr n (n-9*((n-1) `div` 9))
  +
forM_ [2 .. m-1] $ \x ->
  +
forM_ [2 .. m`div`n-1] $ \n ->
  +
incArray arr x n
  +
return arr
  +
)
  +
where
  +
m=10^6
  +
incArray arr x n = do
  +
a <- readArray arr x
  +
b <- readArray arr n
  +
ab <- readArray arr (x*n)
  +
when(ab<a+b) (writeArray arr (x*n) (a + b))
  +
writ x=appendFile "p159.log"$ show x ++ "\n"
  +
main=mapM_ writ $U.elems spfArray
  +
problem_159 = main
  +
  +
--at first ,make main to get file "p159.log"
  +
--then ,add all num in the file
  +
</haskell>
  +
  +
== [http://projecteuler.net/index.php?section=problems&id=160 Problem 160] ==
  +
Factorial trailing digits
  +
  +
We use the following two facts:
  +
  +
Fact 1: <hask>(2^(d + 4*5^(d-1)) - 2^d) `mod` 10^d == 0</hask>
  +
  +
Fact 2: <hask>product [n | n <- [0..10^d], gcd n 10 == 1] `mod` 10^d == 1</hask>
  +
  +
We really only need these two facts for the special case of
  +
<hask>d == 5</hask>, and we can verify that directly by
  +
evaluating the above two Haskell expressions.
  +
  +
More generally:
  +
  +
Fact 1 follows from the fact that the group of invertible elements
  +
of the ring of integers modulo <hask>5^d</hask> has
  +
<hask>4*5^(d-1)</hask> elements.
  +
  +
Fact 2 follows from the fact that the group of invertible elements
  +
of the ring of integers modulo <hask>10^d</hask> is isomorphic to the product
  +
of a cyclic group of order 2 and another cyclic group.
  +
  +
Solution:
  +
<haskell>
  +
problem_160 = trailingFactorialDigits 5 (10^12)
  +
  +
trailingFactorialDigits d n = twos `times` odds
  +
where
  +
base = 10 ^ d
  +
x `times` y = (x * y) `mod` base
  +
multiply = foldl' times 1
  +
x `toPower` k = multiply $ genericReplicate n x
  +
e = facFactors 2 n - facFactors 5 n
  +
twos
  +
| e <= d = 2 `toPower` e
  +
| otherwise = 2 `toPower` (d + (e - d) `mod` (4 * 5 ^ (d - 1)))
  +
odds = multiply [odd | a <- takeWhile (<= n) $ iterate (* 2) 1,
  +
b <- takeWhile (<= n) $ iterate (* 5) a,
  +
odd <- [3, 5 .. n `div` b `mod` base],
  +
odd `mod` 5 /= 0]
  +
  +
-- The number of factors of the prime p in n!
  +
facFactors p = sum . zipWith (*) (iterate (\x -> p * x + 1) 1) .
  +
tail . radix p
  +
  +
-- The digits of n in base b representation
  +
radix p = map snd . takeWhile (/= (0, 0)) .
  +
iterate ((`divMod` p) . fst) . (`divMod` p)
  +
</haskell>
  +
it have another fast way to do this .
  +
  +
Solution:
  +
<haskell>
  +
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)
  +
  +
productMod =foldl (mulMod (10^5)) 1
  +
hFacial 0=1
  +
hFacial a
  +
|gcd a 5==1=(a*hFacial(a-1)) `mod` (5^5)
  +
|otherwise=hFacial(a-1)
  +
fastFacial a= hFacial $a `mod` 6250
  +
numPrime x p=takeWhile(>0) [x `div` (p^a)|a<-[1..]]
  +
p160 x=mulMod t5 a b
  +
where
  +
t5=10^5
  +
lst=numPrime x 5
  +
a=powMod t5 1563 $c `mod` 2500
  +
b=productMod c6
  +
c=sum lst
  +
c6=map fastFacial $x:lst
  +
problem_160 = p160 (10^12)
  +
  +
</haskell>

Latest revision as of 08:22, 23 February 2010

Contents

[edit] 1 Problem 151

Paper sheets of standard sizes: an expected-value problem.

Solution:

problem_151 = fun (1,1,1,1)
 
fun (0,0,0,1) = 0
fun (0,0,1,0) = fun (0,0,0,1) + 1
fun (0,1,0,0) = fun (0,0,1,1) + 1
fun (1,0,0,0) = fun (0,1,1,1) + 1
fun (a,b,c,d) = 
    (pickA + pickB + pickC + pickD) / (a + b + c + d)
    where
    pickA | a > 0 = a * fun (a-1,b+1,c+1,d+1)
          | otherwise = 0
    pickB | b > 0 = b * fun (a,b-1,c+1,d+1)
          | otherwise = 0
    pickC | c > 0 = c * fun (a,b,c-1,d+1)
          | otherwise = 0
    pickD | d > 0 = d * fun (a,b,c,d-1)
          | otherwise = 0

[edit] 2 Problem 152

Writing 1/2 as a sum of inverse squares

Note that if p is an odd prime, the sum of inverse squares of all terms divisible by p must have reduced denominator not divisible by p.

Solution:

import Data.Ratio
import Data.List
import Data.Ord (comparing)
import Data.Function (on)
 
invSq n = 1 % (n * n)
sumInvSq = sum . map invSq
 
subsets (x:xs) = let s = subsets xs in s ++ map (x :) s
subsets _      = [[]]
 
primes = 2 : 3 : 7 : [p | p <- [11, 13..79],
                          all (\q -> p `mod` q /= 0) [3, 5, 7]]
 
-- All subsets whose sum of inverse squares,
-- when added to x, does not contain a factor of p
pfree s x p = [(y, t) | t <- subsets s, let y =  x + sumInvSq t,
                        denominator y `mod` p /= 0]
 
 
-- All pairs (x, s) where x is a rational number whose reduced
-- denominator is not divisible by any prime greater than 3;
-- and s is all sets of numbers up to 80 divisible
-- by a prime greater than 3, whose sum of inverse squares is x.
only23 = foldl fun [(0, [[]])] [13, 7, 5]
    where
    fun a p = 
        collect $ [(y, u ++ v) |
        (x, s) <- a,
        (y, v) <- pfree (terms p) x p,
        u <- s]
    terms p = 
        [n * p | 
        n <- [1..80`div`p],
        all (\q -> n `mod` q /= 0) $
        11 : takeWhile (>= p) [13, 7, 5]
        ]
    collect = 
        map (\z -> (fst $ head z, map snd z)) .
        groupBy fstEq . sortBy cmpFst
    fstEq = (==) `on` fst
    cmpFst = comparing fst
 
-- All subsets (of an ordered set) whose sum of inverse squares is x
findInvSq x y = 
    fun x $ zip3 y (map invSq y) (map sumInvSq $ init $ tails y)
    where
    fun 0 _        = [[]]
    fun x ((n, r, s):ns)
        | r > x     = fun x ns
        | s < x     = []
        | otherwise = map (n :) (fun (x - r) ns) ++ fun x ns
    fun _ _        = []
 
-- All numbers up to 80 that are divisible only by the primes
-- 2 and 3 and are not divisible by 32 or 27.
all23 = [n | a <- [0..4], b <- [0..2], let n = 2^a * 3^b, n <= 80]
 
solutions = 
    [sort $ u ++ v |
    (x, s) <- only23,
    u <- findInvSq (1%2 - x) all23,
    v <- s
    ]
 
problem_152 = length solutions

[edit] 3 Problem 153

Investigating Gaussian Integers

[edit] 4 Problem 154

Exploring Pascal's pyramid.

[edit] 5 Problem 155

Counting Capacitor Circuits.

Solution:

--http://www.research.att.com/~njas/sequences/A051389
a051389=
   [1, 2, 4, 8, 20, 42,
    102, 250, 610, 1486, 
    3710, 9228, 23050, 57718,
    145288, 365820, 922194, 2327914
   ]
problem_155 = sum a051389

[edit] 6 Problem 156

Counting Digits

[edit] 7 Problem 157

Solving the diophantine equation 1/a+1/b= p/10n

Solution:

-- Call (a,b,p) a primitive tuple of equation 1/a+1/b=p/10^n
-- a and b are divisors of 10^n, gcd a b == 1, a <= b and a*b <= 10^n
-- I noticed that the number of variants with a primitive tuple
-- is equal to the number of divisors of p.
-- So I produced all possible primitive tuples per 10^n and
-- summed all the number of divisors of every p
 
import Data.List
k `divides` n = n `mod` k == 0
 
divisors n 
    | n == 10 = [1,2,5,10]
    | otherwise = 
        [ d |
        d <- [1..n `div` 5],
        d `divides` n ] 
        ++ [n `div` 4, n `div` 2,n]
fp n = 
    [ n*(a+b) `div` ab |
    a <- ds,
    b <- dropWhile (<a) ds,
    gcd a b == 1,
    let ab = a*b,
    ab <= n 
    ]
    where
    ds = divisors n
numDivisors :: Integer -> Integer
numDivisors n = product [ toInteger (a+1) | (p,a) <- primePowerFactors n]
numVgln = sum . map numDivisors . fp
 
main = do
    print . sum . map numVgln . takeWhile (<=10^9) . iterate (10*) $ 10 
primePowerFactors x = [(head a ,length a)|a<-group$primeFactors x] 
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..]]
primeFactors n =
    factor n primes
    where
    factor n (p:ps) 
        | p*p > n        = [n]
        | n `mod` p == 0 = p : factor (n `div` p) (p:ps)
        | otherwise      = factor n ps

[edit] 8 Problem 158

Exploring strings for which only one character comes lexicographically after its neighbour to the left.

Solution:

factorial n = product [1..toInteger n]
fallingFactorial x n = product [x - i | i <- [0..fromIntegral n - 1] ]
choose n k = fallingFactorial n k `div` factorial k
fun n=(2 ^ n - n - 1) * choose 26 n 
problem_158=maximum$map fun [1..26]

[edit] 9 Problem 159

Digital root sums of factorisations.

Solution:

import Control.Monad
import Data.Array.ST
import qualified Data.Array.Unboxed as U
spfArray :: U.UArray Int Int
spfArray  = runSTUArray (do
  arr <- newArray (2,m-1) 0 
  forM_ [2 .. m-1] $ \n ->
    writeArray arr n (n-9*((n-1) `div` 9))
  forM_ [2 .. m-1] $ \x ->
    forM_ [2 .. m`div`n-1] $ \n ->
      incArray arr x n
  return arr 
  )
  where
  m=10^6
  incArray arr x n = do
      a <- readArray arr x
      b <- readArray arr n
      ab <- readArray arr (x*n)
      when(ab<a+b) (writeArray arr (x*n) (a + b))
writ x=appendFile "p159.log"$ show x ++ "\n"
main=mapM_ writ $U.elems spfArray
problem_159 = main
 
--at first ,make main to get file "p159.log"
--then ,add all num in the file

[edit] 10 Problem 160

Factorial trailing digits

We use the following two facts:

Fact 1:
(2^(d + 4*5^(d-1)) - 2^d) `mod` 10^d == 0
Fact 2:
product [n | n <- [0..10^d], gcd n 10 == 1] `mod` 10^d == 1

We really only need these two facts for the special case of

d == 5
, and we can verify that directly by

evaluating the above two Haskell expressions.

More generally:

Fact 1 follows from the fact that the group of invertible elements

of the ring of integers modulo
5^d
has
4*5^(d-1)
elements.

Fact 2 follows from the fact that the group of invertible elements

of the ring of integers modulo
10^d
is isomorphic to the product

of a cyclic group of order 2 and another cyclic group.

Solution:

problem_160 = trailingFactorialDigits 5 (10^12)
 
trailingFactorialDigits d n = twos `times` odds
  where
    base = 10 ^ d
    x `times` y = (x * y) `mod` base
    multiply = foldl' times 1
    x `toPower` k = multiply $ genericReplicate n x
    e = facFactors 2 n - facFactors 5 n
    twos
     | e <= d    = 2 `toPower` e
     | otherwise = 2 `toPower` (d + (e - d) `mod` (4 * 5 ^ (d - 1)))
    odds = multiply [odd | a <- takeWhile (<= n) $ iterate (* 2) 1,
                           b <- takeWhile (<= n) $ iterate (* 5) a,
                           odd <- [3, 5 .. n `div` b `mod` base],
                           odd `mod` 5 /= 0]
 
-- The number of factors of the prime p in n!
facFactors p = sum . zipWith (*) (iterate (\x -> p * x + 1) 1) .
               tail . radix p
 
-- The digits of n in base b representation
radix p = map snd . takeWhile (/= (0, 0)) .
          iterate ((`divMod` p) . fst) . (`divMod` p)

it have another fast way to do this .

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)
 
productMod =foldl (mulMod (10^5)) 1
hFacial 0=1
hFacial a
    |gcd a 5==1=(a*hFacial(a-1)) `mod` (5^5)
    |otherwise=hFacial(a-1)
fastFacial a= hFacial $a `mod` 6250
numPrime x p=takeWhile(>0) [x `div` (p^a)|a<-[1..]]
p160 x=mulMod t5 a b
    where
    t5=10^5
    lst=numPrime x 5
    a=powMod t5 1563 $c `mod` 2500
    b=productMod  c6 
    c=sum lst
    c6=map fastFacial $x:lst
problem_160 = p160 (10^12)