# Difference between revisions of "Euler problems/31 to 40"

m (Corrected the links to the Euler project) |
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Solution: |
Solution: |
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<haskell> |
<haskell> |
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− | problem_34 = undefined |
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+ | import Data.Char |
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+ | problem_34 = sum [ x | x <- [3..100000], x == facsum x ] |
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+ | where facsum = sum . map (product . enumFromTo 1 . digitToInt) . show |
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</haskell> |
</haskell> |
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| otherwise = Nothing] |
| otherwise = Nothing] |
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where isPan s = sort s == "123456789" |
where isPan s = sort s == "123456789" |
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+ | </haskell> |
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+ | |||

+ | Other solution: |
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+ | <haskell> |
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+ | import Data.List |
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+ | |||

+ | mult n i vs | length (concat vs) >= 9 = concat vs |
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+ | | otherwise = mult n (i+1) (vs ++ [show (n * i)]) |
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+ | |||

+ | problem_38 :: Int |
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+ | problem_38 = maximum $ map read $ filter ((['1'..'9'] ==) .sort) $ [ mult n 1 [] | n <- [2..9999] ] |
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</haskell> |
</haskell> |
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## Revision as of 12:28, 5 August 2007

## Contents

## 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:

```
import Data.Char
problem_34 = sum [ x | x <- [3..100000], x == facsum x ]
where facsum = sum . map (product . enumFromTo 1 . digitToInt) . show
```

## 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"
```

Other solution:

```
import Data.List
mult n i vs | length (concat vs) >= 9 = concat vs
| otherwise = mult n (i+1) (vs ++ [show (n * i)])
problem_38 :: Int
problem_38 = maximum $ map read $ filter ((['1'..'9'] ==) .sort) $ [ mult n 1 [] | n <- [2..9999] ]
```

## 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))
```