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

BrettGiles (talk | contribs) m (EulerProblems/31 to 40 moved to Euler problems/31 to 40) |
(→[http://projecteuler.net/index.php?section=problems&id=39 Problem 39]: a solution (could use some cleanup)) |
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Line 75: | Line 75: | ||

Solution: |
Solution: |
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+ | 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. |
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<haskell> |
<haskell> |
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− | problem_39 = |
+ | problem_39 = head $ perims !! indexMax |

+ | where perims = group |
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+ | $ sort [n*p | p <- pTriples, n <- [1..1000 `div` p]] |
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+ | counts = map length perims |
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+ | Just indexMax = findIndex (== (maximum counts)) $ counts |
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+ | pTriples = [p | |
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+ | n <- [1..250], |
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+ | m <- [n+1..250], |
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+ | even n || even m, |
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+ | gcd n m == 1, |
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+ | let a = m^2 - n^2, |
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+ | let b = 2*m*n, |
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+ | let c = m^2 + n^2, |
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+ | let p = a + b + c, |
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+ | p < 1000] |
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</haskell> |
</haskell> |
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## Revision as of 04:14, 30 March 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:

```
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 = undefined
```

## 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..250],
m <- [n+1..250],
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 = undefined
```