Find the smallest number that can be expressed as
the sum of 3 consecutive prime numbers,
the sum of 29 consecutive prime numbers,
the sum of 373 consecutive prime numbers,
the sum of 665 consecutive prime numbers,
and is itself a prime number.

The idea behind my solution is that if I have four lists of the sums, and a list of primes, then the solution to the problem is the smallest member of the intersection of all five lists.

I saved the problem of finding primes to the last and instead concentrated on summing up lists of integers in a convenient way. I ended up with the following solution to that:

sumN :: Int -> [Integer]
sumN n = map fst sumi
    where
        sumi = tail $ iterate (sumn) (0, primes)
        sumn (r, xs) = (sum $ take n xs, tail xs)

The real work happens in the inner functions. sumn picks out as many numbers as I want and sums them up, the reason for taking and returning a tuple is that sumi can use iterate to calculate the sequence of sums. The first item in the result of iterate a is a, so sumi drops the first one. This lets me create the needed lists of sums:

sum3 = sumN 3
sum29 = sumN 29
sum373 = sumN 373
sum665 = sumN 665

After having a quick look at the implementation of Data.List.intersect I realised it probably would be a bit too slow. Instead I wrote one that exploits that all the involved lists are ordered:

intersection a@(x:xs) b@(y:ys)
    | x == y = x : intersection xs ys
    | x < y = intersection (dropWhile (< y) xs) b
    | x > y = intersection a (dropWhile (x >) ys)

After convincing myself that this code did what I wanted (basically through playing in GHCi using an incorrect definition of primes, primes = [1..]) I turned to the problem of finding the primes. Of course someone else had already written the Haskell code to find primes. That code fitted perfectly with mine since it produces an infinate list of Integers. Brilliant!

Finally here’s the main that allows me to compile the code and get the answer quickly (it’s slightly modified for the medium, basically I wrote it all on one, long line):

main :: IO ()
main = let
        i1 = (primes `intersection` sum665)
        i2 = (sum3 `intersection` sum373)
        s =  i1 `intersection` i2 `intersection` sum29
    in
        print $ head s

Compiled, it runs in about 5.5 seconds on my desktop machine.

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The second, summing certain lines taken from files matching certain glob patterns, was easily solved with a short Haskell program.

The third, a routing problem was probably the easiest and I think most easily solved on paper. I did make a mistake though, which I caught before typing my answer into the webform.

The fourth one, adding sequences of primes, was fairly straight forward. I think it would have presented a bit more trouble if I didn’t make use of Haskell’s laziness and support for infinate lists though.

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http://www.io.com/~jsm/nph-ap.cgi/http://www.scroogle.org/cgi-bin/nbbw.cgi?Gw=%s&n=2

Now, some things to ponder for the paranoid out there:

• Is Scroogle not evil? Does it log your IP?
• What logging does AutoPOST do?

[Edited 06-10-2006 17:46 BST] It turns out Scroogle already offers this.

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• Calculator (2 * 2 + 2, 45% of 60)
• List of definitions (define:google)
• Unit converter (85 kg in lbs)
• and much more…

Google is amazingly useful!

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