Mon Jun 29 14:36:11 UTC 2020

```Without having attempted to code this in any particular language, but
just thinking about the problem, I believe the CSES knights problem is
not a test of language speed or programming acumen but a test of
choosing an efficient choice of algorithm that doesn't generate more
information than the question asks for.

In short, they're asking for the NUMBER of boards, not the actual
boards. Most of the solutions people are proposing in Haskell simulate
the problem instead of calculating it -- in short, they generate the
actual boards, then count them, whereas the solution only requires you
to count them.

The solutions for n = 1 and n = 2 can be calculated by hand and put in
as constants. n = 3 you can calculate or simulate as is your preference.

Knights have a maximum interaction range of three linear squares.
Knights placed more than three squares apart in any one direction cannot
hinder each other. So the number of illegal placements of knights can be
confined to a 3x3 board. After that, all illegal boards of size n x n
are simply one of those 3 x 3 boards shifted to a new position.

The total number of n x n boards with two knights placed on them is
given by (n^2) choose 2, which I'm not going to look up because it's
been over 20 years since I took statistics and I'm happy about that.
Still, it's a calculation. Not a super simple one from a computing
perspective, since it involved factorials, but I'm assuming someone has
figured out how to do factorials quickly in Haskell?

The number of places you can position a 3 x 3 board within an n x n
space is something like (n-3)^2 if I'm not mistaken?

So you can subtract that from the total number of boards to arrive at a
result.

NOTE: This likely requires SOME tweaking for edge cases (For example, Is
the board where k1 is in position x and k2 is in position y considered
the same as the board where their positions reversed, or not? Does the
choosing calculation factor for that properly?) because it's literally a
ten-second-in-the-shower concept, but it seems like this could come up
with a result. Whether it does it in time is mostly down to whether
Haskell can do factorials fast enough at that point.

On 2020-06-28 8:50 p.m., Julian Ong wrote:
> After the Two Knights problem, I went on this next problem which
> requires that you separate 1..n into two sets with the same sum if
> possible. Again my algorithm in Haskell works but is apparently too
> slow. It fails for CSES test inputs >= 26560 where a solution exists.
>
> I'm starting to wonder if Haskell is fundamentally too slow compared
> to other languages. From what I've read that shouldn't be the case
> though. For this problem it looks like it's doable in Python (I
> haven't tried that). Most of the fastest solutions for these problems
> seem to be written in C++. If there's anyone who's trying to solve
> these problems in Haskell (they're really fun by the way if you've
> never checked them out) and has solved this one (or Two Knights) and
> passed all the tests, I'd love to hear how you did it. Thanks.
>
> ---
>
> -- CSES - Two Sets
> -- Given 1..n, separate into two sets of equal sums and if possible
> list the elements of each set of a possible solution or output NO if
> not possible
>
> main :: IO ()
> main = do
>     line <- getLine
>     let n = read line :: Integer
>     putStrLn \$ solveN n
> -- Observe that sum [1..n] = n*(n+1)/2 so each set sum must be
> n*(n+1)/4 and so the set sum must be divisible by 4 for the separation
> to be possible
> -- Then the algorithm starts adding numbers from n down to 1 until the
> next number would make the sum exceed the required set sum
> -- At this point you add one more number, which will be the lowest
> number, to fill in the gap to complete set1. Set2 is then the other
> numbers.
> solveN :: Integer -> String
> solveN n
>     | (n * (n+1) `mod` 4 /= 0) = "NO"
>     | otherwise = "YES\n" ++ show set1_count ++ "\n" ++ (unwords \$ map
> show set1_list) ++ "\n" ++ show set2_count ++ "\n" ++ (unwords \$ map
> show set2_list)
>         where
>             set_sum = (n * (n+1)) `div` 4
>             set1_part1 = takeWhile (\x -> x*(n+1-x) + sum [0..(n-x)] <
> (n * (n+1)) `div` 4) [n, n-1..1]
>             set1_part2 = set_sum - sum set1_part1
>             set1_list = set1_part1 ++ [set1_part2]
>             set1_count = (toInteger . length) set1_list
>             set2_list = [x | x <- [1..n], not (x `elem` set1_list)]
>             set2_count = (toInteger . length) set2_list
> ----
>
> Julian
>
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