# Euler problems/31 to 40

### From HaskellWiki

(→[http://projecteuler.net/index.php?section=problems&id=39 Problem 39]: a solution (could use some cleanup)) |
(→[http://projecteuler.net/index.php?section=problems&id=39 Problem 39]: reduce search space) |
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Line 83: | Line 83: | ||

Just indexMax = findIndex (== (maximum counts)) $ counts | Just indexMax = findIndex (== (maximum counts)) $ counts | ||

pTriples = [p | | pTriples = [p | | ||

− | n <- [1.. | + | n <- [1..floor (sqrt 1000)], |

− | m <- [n+1.. | + | m <- [n+1..floor (sqrt 1000)], |

even n || even m, | even n || even m, | ||

gcd n m == 1, | gcd n m == 1, |

## Revision as of 08:39, 30 March 2007

## Contents |

## 1 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

## 2 Problem 32

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

Solution:

problem_32 = undefined

## 3 Problem 33

Discover all the fractions with an unorthodox cancelling method.

Solution:

problem_33 = undefined

## 4 Problem 34

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

Solution:

problem_34 = undefined

## 5 Problem 35

How many circular primes are there below one million?

Solution:

problem_35 = undefined

## 6 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

## 7 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

## 8 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 = undefined

## 9 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]

## 10 Problem 40

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

Solution:

problem_40 = undefined