{-
    Number of combinations of a NxN Rubik's cube and variations.

    Copyright 2016 Ken Takusagawa

    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with this program.  If not, see <http://www.gnu.org/licenses/>.

 -}

{-# LANGUAGE ScopedTypeVariables, LambdaCase #-}
module Main where {
import System.Environment(getArgs);
import Data.Function((&));

import Math.Combinatorics.Exact.Factorial(factorial);

main :: IO();
main = getArgs >>= \case{
["cube",n] -> read n & cube & print;
["supercube",n] -> read n & supercube & print;
["supersupercube",n] -> read n & super_supercube & print;
["megaminx",n] -> read n & megaminx & print;
_ -> error "bad arguments";
};

-- http://www.speedcubing.com/chris/cubecombos.html

div_exact :: Integer -> Integer -> Integer;
div_exact x y = case divMod x y of {
(q,0) -> q;
_ -> error "division had a remainder";
};

common :: Integer -> Integer -> Integer -> Integer;
common i den_base n = let {
e1 :: Integer;
e1 = div (n*n - 2*n) 4;
e2 :: Integer;
e2 = div ((n-2)^2) 4;
numerator :: Integer;
numerator = (24 * 2^i * factorial 12)^(mod n 2) * factorial 7 * 3^6 * (factorial 24)^e1;
denominator :: Integer;
denominator = den_base ^ e2;
} in div_exact numerator denominator;

-- https://oeis.org/A075152
cube :: Integer -> Integer;
cube = common 10 ((factorial 4)^6);

-- https://oeis.org/A080660
supercube :: Integer -> Integer;
supercube = common 21 2;

-- https://oeis.org/A080659
super_supercube :: Integer -> Integer;
super_supercube n = let {
e1 :: Integer;
e1 = div_exact (n^3-4*n+(mod n 2)*(12 - 9*n)) 24;
e2 :: Integer;
e2 = div (n-2) 2 + mod n 2;
e4 :: Integer;
e4 = mod n 2 * div n 2;
} in (div_exact (factorial 24) 2)^e1 * 24^e2 * (factorial 7 * 3^6)^(div n 2) * (factorial 12 * 2^21)^e4;

-- http://michael-gottlieb.blogspot.com/2008/05/number-of-positions-of-generalized.html
-- n=1 megaminx; n=2 gigaminx; etc.
megaminx :: Integer -> Integer;
megaminx n = div_exact (factorial 30 * factorial 20 * (factorial 60)^(n^2-1) * (factorial 5)^(12*n) * 2^28 * 3^19) ((factorial 5)^(12*n^2) * 2^n);
-- agrees with results of http://twistypuzzles.com/~sandy/forum/viewtopic.php?f=1&t=24058

} --end