Euler problems/31 to 40

Problem 31

Investigating combinations of English currency denominations.

Solution:

This is the naive doubly recursive solution. Speed would be greatly improved by use of memoization, dynamic programming, or the closed form.

problem_31 = pence 200 [1,2,5,10,20,50,100,200]
    where pence 0 _  = 1
          pence n [] = 0
          pence n denominations@(d:ds)
                | n < d     = 0
                | otherwise = pence (n - d) denominations
                              + pence n ds

Problem 32

Find the sum of all numbers that can be written as pandigital products.

Solution:

problem_32 = undefined

Problem 33

Discover all the fractions with an unorthodox cancelling method.

Solution:

problem_33 = undefined

Problem 34

Find the sum of all numbers which are equal to the sum of the factorial of their digits.

Solution:

problem_34 = undefined

Problem 35

How many circular primes are there below one million?

Solution:

problem_35 = undefined

Problem 36

Find the sum of all numbers less than one million, which are palindromic in base 10 and base 2.

Solution:

problem_36 = undefined

Problem 37

Find the sum of all eleven primes that are both truncatable from left to right and right to left.

Solution:

problem_37 = undefined

Problem 38

What is the largest 1 to 9 pandigital that can be formed by multiplying a fixed number by 1, 2, 3, ... ?

Solution:

problem_38 = maximum $ catMaybes [result | j <- [1..9999],
              let p2 = show j ++ show (2*j),
              let p3 = p2 ++ show (3*j),
              let p4 = p3 ++ show (4*j),
              let p5 = p4 ++ show (5*j),
              let result
                      | isPan p2 = Just p2
                      | isPan p3 = Just p3
                      | isPan p4 = Just p4
                      | isPan p5 = Just p5
                      | otherwise = Nothing]
  where isPan s = sort s == "123456789"

Problem 39

If p is the perimeter of a right angle triangle, {a, b, c}, which value, for p ≤ 1000, has the most solutions?

Solution: We use the well known formula to generate primitive Pythagorean triples. All we need are the perimeters, and they have to be scaled to produce all triples in the problem space.

problem_39 = head $ perims !! indexMax
    where  perims = group
                    $ sort [n*p | p <- pTriples, n <- [1..1000 `div` p]]
           counts = map length perims
           Just indexMax = findIndex (== (maximum counts)) $ counts
           pTriples = [p |
                       n <- [1..floor (sqrt 1000)],
                       m <- [n+1..floor (sqrt 1000)],
                       even n || even m,
                       gcd n m == 1,
                       let a = m^2 - n^2,
                       let b = 2*m*n,
                       let c = m^2 + n^2,
                       let p = a + b + c,
                       p < 1000]

Problem 40

Finding the nth digit of the fractional part of the irrational number.

Solution:

problem_40 = (d 1)*(d 10)*(d 100)*(d 1000)*(d 10000)*(d 100000)*(d 1000000)
    where n = concat [show n | n <- [1..]]
          d j = Data.Char.digitToInt (n !! (j-1))