# Rubiks Cube

## Rubik's Cube represented with faces

Here is a simple model for a Rubik's Cube.

The basic idea is that you only need to keep track of the corners and edges. Each corner has three faces. Each edge has two faces. Keeping track of a face means telling where it was before any moves were made and where it is in the current state.

Choose, as a convention, the ordering, right, up, front. (Math/Physics folk: this is in anology to the "right hand rule" convention which assigns an ordering to the "x y and z" axes and determines that z will be "out of" rather than "into" the plane)

For example, the lower left front corner would be represented as (Left Left) (Down Down) (Front Front) before any moves are made Then, after a rotation about the Front face, the same corner, now in the right down front position would be represented as (Left Down) (Down Right) (Front Front).

Edit by somebody else: I'm not the author of this, however I think there are some erros in the definition of all the datas below. I think they all are missing the constructor, when you're reading the code keep that in mind. also:

You can read the paper by Richard E. Korf named "Finding Optimal Solutions to Rubik's Cube Using Pattern Databases." to have a better understanding of the Edged/Corners approach

```#!/usr/bin/runhugs
module Main (main) where
main                    :: IO ()
main =  do putStr "Not your ordinary language"

data Cube = Edges Corners

type Edges = [Edge]
-- or Edges = Edge Edge Edge Edge Edge Edge Edge Edge Edge Edge Edge Edge

type Corners = [Corner]

data Edge = Face Face

data Corner = Face Face Face

data Face = Was Is

data Was = R|L|U|D|F|B
data Is =  R|L|U|D|F|B
```

## Rubik's Cube as space transformation, with a solver

by Péter Diviánszky

A Rubik's Cube configuration can be represented by telling how to transform the space such that the 27 small cubes in the original configuration are moved and rotated to make up the current configuration.

```type CubeFun = (Int, Int, Int) -> (Int, Int, Int)
```

The identity function represents the original cube.

The mid points of the 27 small cubes are (x,y,z) where x,y,z ∈ {-10,0,10}. The magnification by 10 is needed because I wanted to avoid floating point calculation. The smaller lengths are used to explore how the space transformation alters directions around the mid points of the small cubes.

The colors are represented with space directions. For example, the direction (1,0,0) is the color of the face of the original cube in that direction. This direction is called 'x'. The color of the opposite face is represented by (-1,0,0) which is called 'X'. 'y', 'Y', 'z', 'Z' correspond to the other colors, respectively.

The rotations are also represented with space directions. For example 'x' represents one quarter move of the top face around the 'x' axis in positive direction. 'X' represents one quarter move of the opposite face around the 'x' axis in negative direction. 'y', 'Y', 'z', 'Z' correspond to other rotations, respectively.

Here is the full source code implementing the moves, a solver, a reader and a pretty printer:

```import Data.List
import Data.Maybe
import System.Environment

div10 x = (x + 5) `div` 10
mod10 x = (x + 5) `mod` 10 - 5

map3 f (x, y, z) = (f x, f y, f z)
zipW3 (*) (x, y, z) (x', y', z') = (x * x', y * y', z * z')
inner p q = x + y + z where (x, y, z) = zipW3 (*) p q
rot3 [a, b, c] p = map3 (inner p . dirVec) (a, b, c)

dirVec 'x' = ( 1, 0, 0)
dirVec 'y' = ( 0, 1, 0)
dirVec 'z' = ( 0, 0, 1)
dirVec 'X' = (-1, 0, 0)
dirVec 'Y' = ( 0,-1, 0)
dirVec 'Z' = ( 0, 0,-1)
dirVec '-' = ( 0, 0, 0)

type CubeFun = (Int, Int, Int) -> (Int, Int, Int)

cubeAt p x | map3 div10 x == p = x
cubeAt _ x = map3 ((10*) . div10) x

faces = map words [" -  Zyx  -   - "
,"yXz xyz Yxz XYz"
," -  zyX  -   - "]

readCube s p = foldr1 (zipW3 (+))
[ map3 (* if dirVec q == map3 mod10 p then 11 else 10) \$
dirVec \$ lines s !! (4*l - y + 1) !! (7*c + 2*(x + 1))
| (l, face) <- zip [2,1,0] faces
, (c, m@[q,_,_]) <- zip [2,1,0,3] face
, (1, x, y) <- [rot3 m \$ map3 div10 p] ]

showCube :: CubeFun -> String
showCube c = unlines \$ map (unlines . map unwords . transpose . map face) faces
where
face "-" = replicate 3 "      "
face f = [ unwords [ filter ((== map3 mod10 (c \$ rot3 f (11, x, y))) . dirVec) "xyzXYZ-"
| x <- [-10,0,10]] ++ " "
| y <- [10,0,-10]]

type Rotation = Char

decodeCube :: [Rotation] -> CubeFun
decodeCube = foldr (.) id . map rot
where
rot d p | div10 (dirVec d `inner` p) <= 0 = p
rot d p = rot3 (fromJust \$ lookup d \$ zip "xyzXYZ" \$ words "xZy zyX Yxz xzY Zyx yXz") p

encodeCube :: CubeFun -> Maybe [Rotation]
encodeCube cfun = listToMaybe \$ filter (\c -> showCube (cfun . decodeCube c) == showCube id)
\$ foldl (flip \$ concatMap . solvePiece) [""] \$ map unzip \$ init \$ tails
[(( 1,-1, 1),"x")
,(( 1,-1,-1),"Z")
,(( 1, 1, 1),"y")
,(( 1, 1,-1),"yyzzxxyyXXzzzXXyyxxzzyZyyyZZyyyZyyy")
,(( 0, 1,-1),"yyZZyyZZzxxYYZYYzzzYzzYYYxxZZyXX")
,(( 0, 1, 1),"zzzYYxxyyZXXyyxxzz")
,((-1,-1,-1),"X")
,(( 1, 1, 0),"zxxZzXXZZYXXzzxxyyyZZZ")
,((-1, 1,-1),"ZZzxxyyzyyZZZxxYYzXzzzXXXYYYzzXXYYYXXZZZzzz")
,(( 0,-1,-1),"ZZyyzzzYYyyZyyzzyyyZYYyyzzzy")
,(( 1, 0,-1),"zzzyyXXyyzzYYxxyyZZZyyyZYYyyzzzy")
,((-1,-1, 1),"XXZXXZZZyyzyyyzzzXXzzzXXzyyy zzxxZZZYYzXXZZZXXYYzzzxxxyyxxxZzzzyXXyyyzzz")
,((-1,-1, 0),"yyxxZZyyzzXYYzzyyxxxyy")
,((-1, 0,-1),"YYZZZXXZYYyyzzzxxzYYyyyZYYyyzzzy")
,((-1, 1, 0),"ZZZYYXXyyzzYYxxZZZzzzyxxzzxxyz")
,(( 0,-1, 1),"xxzzzyyXXYYZYYXXyyzzxx")
,(( 1,-1, 0),"ZyyxxYYZZYYxxZZZzzzyzzxxzzyzzz")
,(( 1, 0, 1),"YyyyxYyyyZYyyyXXYYYyZYYYyxYYYyzz")]
where
solvePiece (fixed: _, moves) p = try [] [""]
where
try _ [] = []
try acc (p': cs)
| sc `elem` acc = try acc cs
| sc == showCube (cubeAt fixed) = [p' ++ p]
| otherwise = try (sc: acc) \$ cs ++ map (p' ++) (moves >>= words)
where
sc = showCube \$ cubeAt fixed . decodeCube p' . decodeCube p . cfun

main = do
args <- getArgs
c <- case args of
[s] | all (`elem` "xyzXYZ") s -> pure \$ decodeCube s
putStr \$ showCube c
putStrLn \$ maybe "not solvable" ("solution: " ++) \$ encodeCube c
```

A sample input file:

```       Z x x
x z y
z Y Y

X Y x  y X Z  x Z Z  y y Y
z Y Y  Z x X  z y z  x X y
x x Y  X y Z  y Z X  y Y z

z X X
Z Z X
Y z z
```

The corresponding output (the solution is not normalized):

```solution: YyyyxYyyyZYyyyXXYYYyZYYYyxYYYyzzxxzzzyyXXYYZYYXXyyzzxxZZZYYXXyyzzYYxxZZZzzzyxxzzxxyzZZZYYXXyyzzYYxxZZZzzzyxxzzxxyzZyyxxYYZZYYxxZZZzzzyzzxxzzyzzzYYZZZXXZYYyyzzzxxzYYyyyZYYyyzzzyxxzzzyyXXYYZYYXXyyzzxxyyxxZZyyzzXYYzzyyxxxyyYYZZZXXZYYyyzzzxxzYYyyyZYYyyzzzyXXZXXZZZyyzyyyzzzXXzzzXXzyyyzzzyyXXyyzzYYxxyyZZZyyyZYYyyzzzyZZyyzzzYYyyZyyzzyyyZYYyyzzzyyyxxZZyyzzXYYzzyyxxxyyyyxxZZyyzzXYYzzyyxxxyyZZzxxyyzyyZZZxxYYzXzzzXXXYYYzzXXYYYXXZZZzzzXXZXXZZZyyzyyyzzzXXzzzXXzyyyXXZXXZZZyyzyyyzzzXXzzzXXzyyyzxxZzXXZZYXXzzxxyyyZZZZZzxxyyzyyZZZxxYYzXzzzXXXYYYzzXXYYYXXZZZzzzZZzxxyyzyyZZZxxYYzXzzzXXXYYYzzXXYYYXXZZZzzzXZZzxxyyzyyZZZxxYYzXzzzXXXYYYzzXXYYYXXZZZzzzzzzYYxxyyZXXyyxxzzXzxxZzXXZZYXXzzxxyyyZZZyyZZyyZZzxxYYZYYzzzYzzYYYxxZZyXXzxxZzXXZZYXXzzxxyyyZZZXyyzzxxyyXXzzzXXyyxxzzyZyyyZZyyyZyyyyyzzxxyyXXzzzXXyyxxzzyZyyyZZyyyZyyyyyZZyyZZzxxYYZYYzzzYzzYYYxxZZyXXyZZyyzzxxyyXXzzzXXyyxxzzyZyyyZZyyyZyyyXxyyyzzxxyyXXzzzXXyyxxzzyZyyyZZyyyZyyyy
```