[vcppbmdn] Rubik's cube combinations

We implement in Haskell the formulae at http://www.speedcubing.com/chris/cubecombos.html to compute the number of combinations of a NxN Rubik's cube, as well as a supercube (for which center facet orientation matters) and a super-supercube (also called a "real" cube, for which inner cubie orientations also matter) variations.

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

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;

Incidentally, the output numbers factor easily. The largest prime factor in them is 23. One could do the calculations over a factor base of primes from 2 to 23 to get the output directly in compact factored form.

Next, we consider the Megaminx, and higher odd-order variants (so not Flowerminx / Kilominx):

-- 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);


Full Haskell source code.