# XQuery/Project Euler

Project Euler is a collection of mathematical problems. Currently there are 166 so it may take some time to get through them all :-).

Add all the natural numbers below 1000 that are multiples of 3 or 5.

sum ((1 to 999)[. mod 3 = 0 or . mod 5 = 0])

Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million.

declare function local:fib($fibs,$max) { let $next := $fibs[1] + $fibs[2] return if ($next > $max) then $fibs else local:fib(($next,$fibs),$max) }; sum( local:fib((2,1),1000000)[. mod 2 = 0])

This brute-force approach recursively builds the Fibonacci sequence (in reverse) up to the maximum, then filters and sums the result.

What is the largest prime factor of the number 317584931803?

First we need to get a list of primes. The algorithm known as the Sieve of Eratosthenes is directly expressible in XQuery:

declare function local:sieve($primes as xs:integer*,$nums as xs:integer* ) as xs:integer* { if (exists($nums)) then let $prime := $nums[1] return local:sieve(($primes,$prime), $nums[. mod $prime != 0]) else $primes }; <result> { local:sieve( (), 2 to 1000 ) } </result>

The list of primes starts off empty, the list of numbers starts off with the integers. Each recursive call of **local:sieve** takes the first of the remaining integers as a new prime and reduces the list of integers to those not divisible by the prime. When the list of integers is exhausted, the list of primes is returned.

Factorization of a number N is also easily expressed as the subset of primes which divide N:

declare function local:factor($n as xs:integer ,$primes as xs:integer*) as xs:integer* { $primes[ $n mod . = 0] };

Hence

let $n:= xs:integer(request:get-parameter("n",100)) let $max := xs:integer(ceiling($n div 2)) let $primes := local:sieve((),2 to $max) return <result> { local:factor($n,$primes) } </result>

And the largest is

max (local:factor($n,$primes))

Sadly this elegant method runs out of space and time for integers as large as that in the problem.

Find the largest palindrome made from the product of two 3-digit numbers.

declare function local:palindromic($n as xs:integer) as xs:boolean { let $s := xs:string($n) let $sc := string-to-codepoints($s) let $sr := reverse ($sc) let $r := codepoints-to-string($sr) return $s = $r }; max( (for $i in (100 to 999) for $j in (100 to 999) return $i * $j) [local:palindromic(.)] )

Run [ takes 20 seconds]

What is the difference between the sum of the squares and the square of the sums for integers from 1 to 100?

declare function local:diff-sum($n as xs:integer) as xs:integer) { sum (1 to $n) * sum(1 to $n) - sum( for $i in 1 to $n return $i * $i ) }; local:diff-sum(100)

This nasty brute-force method can be replaced by an explicit expression using familiar formula:

declare function local:diff-sum($n as xs:integer) as xs:integer { let $sum := $n * ($n + 1) div 2 let $sumsq :=( $n * ($n+1) * (2 * $n +1) ) div 6 return $sum * $sum - $sumsq }; local:diff-sum(100)