Em portugues..

main_Main.java

package main;

/**
 * Created by lucas on 22/08/16.
 */
public class Main {

    public static void main(String[] args) {
        new Square().printSquare();
    }

    static long c = 0;

    public static void permutation(String str) {
        permutation("", str);
    }

    private static void permutation(String prefix, String str) {
        int n = str.length();
        if (n == 0) {
            System.out.println(prefix);
        }
        else {
            for (int i = 0; i < n; i++) {
                permutation(prefix + str.charAt(i), str.substring(0, i) + str.substring(i + 1, n));
            }
        }
    }
}

main_Peca.java

package main;

/**
 * Created by lucas on 22/08/16.
 */
public enum Peca {

    A(Type.CANTO, "A"), B(Type.MEIO, "B"), C(Type.CANTO, "C"), D(Type.MEIO, "D"),

    E(Type.CANTO, "E"), F(Type.MEIO, "F"), G(Type.CANTO, "G"), H(Type.MEIO, "H"),


    I(Type.CANTO, "I"), J(Type.MEIO, "J"), K(Type.CANTO, "K"), L(Type.MEIO, "L"),

    M(Type.CANTO, "M"), N(Type.MEIO, "N"), O(Type.CANTO, "O"), P(Type.MEIO, "P");

    private Type tipo;

    private String nome;

    Peca(Type tipo, String name){
        this.tipo = tipo;

        this.nome = name;
    }

    public Type getTipo() {
        return tipo;
    }

    public String getNome() {
        return nome;
    }

    public void setNome(String nome) {
        this.nome = nome;
    }
}

main_Posicao.java

package main;

/**
 * Created by lucas on 22/08/16.
 */
public class Posicao {

    private Peca peca;

    public Posicao(Peca peca){

        this.peca = peca;
    }

    public Peca getPeca() {

        return peca;
    }
}

main_Square.java

package main;

/**
 * Created by lucas on 23/08/16.
 */
public class Square {

    private Peca[] pecas;
    private Posicao[] posicoes;

    public Square() {
        pecas = new Peca[16];
        posicoes = new Posicao[24];

        pecas[0] = Peca.A;/*--*/pecas[8] = Peca.I;

        pecas[1] = Peca.B;/*--*/pecas[9] = Peca.J;

        pecas[2] = Peca.C;/*--*/pecas[10] = Peca.K;

        pecas[3] = Peca.D;/*--*/pecas[11] = Peca.L;

        pecas[4] = Peca.E;/*--*/pecas[12] = Peca.M;

        pecas[5] = Peca.F;/*--*/pecas[13] = Peca.N;

        pecas[6] = Peca.G;/*--*/pecas[14] = Peca.O;

        pecas[7] = Peca.H;/*--*/pecas[15] = Peca.P;


        posicoes[0] = new Posicao(pecas[0]);/*--*/posicoes[12] = new Posicao(pecas[8]);

        posicoes[1] = new Posicao(pecas[0]);/*--*/posicoes[13] = new Posicao(pecas[8]);

        posicoes[2] = new Posicao(pecas[1]);/*--*/posicoes[14] = new Posicao(pecas[9]);

        posicoes[3] = new Posicao(pecas[2]);/*--*/posicoes[15] = new Posicao(pecas[10]);

        posicoes[4] = new Posicao(pecas[2]);/*--*/posicoes[16] = new Posicao(pecas[10]);

        posicoes[5] = new Posicao(pecas[3]);/*--*/posicoes[17] = new Posicao(pecas[11]);

        posicoes[6] = new Posicao(pecas[4]);/*--*/posicoes[18] = new Posicao(pecas[12]);

        posicoes[7] = new Posicao(pecas[4]);/*--*/posicoes[19] = new Posicao(pecas[12]);

        posicoes[8] = new Posicao(pecas[5]);/*--*/posicoes[20] = new Posicao(pecas[13]);

        posicoes[9] = new Posicao(pecas[6]);/*--*/posicoes[21] = new Posicao(pecas[14]);

        posicoes[10] = new Posicao(pecas[6]);/*--*/posicoes[22] = new Posicao(pecas[14]);

        posicoes[11] = new Posicao(pecas[7]);/*--*/posicoes[23] = new Posicao(pecas[15]);
    }

    public void u(int movimento) {
        for (int i = 0; i < 12 - movimento; i++) {

            moverTopo();
        }
    }

    public void d(int movimento) {
        for (int i = 0; i < 12 - movimento; i++) {

            moverBase();
        }
    }

    public void s() {
        Posicao[] x = posicoes;

        //A "slash" permutes indexes 3-8 between
        //14-19. The "indexOfStartPerms" represents
        //the first index to start that permutations,
        //then it will be permuted with his same
        //(index + 11). This cycle of permutations needs
        //to repeat for 6 times.

        int indexDoInicioDasPerms = 3;

        for (int i = indexDoInicioDasPerms; i < 6 + indexDoInicioDasPerms; i++) {

            Posicao aux = x[i];

            x[i] = x[i + 11];

            x[i + 11] = aux;
        }

        posicoes = x;
    }

    public boolean isTravado() {
        //Square is blocked to do a slash if any of this posicoes
        // (indexes) [2,3], [8,9], [13,14] and [19,20] contains, at
        // same time, the same Peca object. If not equals in all
        // cases, this method will return false (not blocked).

        return
                posicoes[2].getPeca().equals(posicoes[3].getPeca()) ||

                posicoes[8].getPeca().equals(posicoes[9].getPeca()) ||

                posicoes[13].getPeca().equals(posicoes[14].getPeca()) ||

                posicoes[19].getPeca().equals(posicoes[20].getPeca());
    }

    public boolean isQuadrado(){
        //This method is able to say if the cube have his
        //booth layers isQuadrado. To make this search, the
        //loop look at the current piece and the next piece
        //of Array. It will return false ("false" for not squared
        //shape) if that pecas have the same type and be
        //different.

        for (int i = 0; i < posicoes.length; i++){

            Peca p1 = posicoes[i].getPeca();

            Peca p2 = posicoes[i+1].getPeca();

            if (p1.getTipo().equals(p2.getTipo()) && !p1.equals(p2)){

                return false;
            }

            if (i == (posicoes.length - 2)){

                break;
            }
        }

        return true;
    }

    private void moverTopo() {
        Posicao[] x = posicoes;

        //(x.length / 2) - 1 -> 11 (indexes 0-11);

        for (int i = 0; i < (x.length / 2) - 1; i++) {

            Posicao aux = x[i];

            x[i] = x[i + 1];

            x[i + 1] = aux;
        }

        posicoes = x;
    }

    private void moverBase() {
        Posicao[] x = posicoes;

        //i = 12 -> bottom pecas starts at index 12 and ends at 23 (x.length - 1);

        for (int i = 12; i < (x.length - 1); i++) {

            Posicao aux = x[i];

            x[i] = x[i + 1];

            x[i + 1] = aux;
        }

        posicoes = x;
    }

    public void printSquare(){
        System.out.println(this);
    }

    @Override
    public String toString() {
        String r = "Topo:\n", r2 = "Topo:\n";

        for (int i = 0; i < posicoes.length; i++) {

            r += "A posicao " + i + " contem a peca: " + posicoes[i].getPeca().getNome() +
                    " (" + posicoes[i].getPeca().getTipo().getTypeName() + ")\n";

            //r2 += posicoes[i].getPeca().getTipo().getTypeName() + ", ";

            r2 += posicoes[i].getPeca().getTipo().getNameAsByte() + ", ";

            if (i == (posicoes.length / 2) - 1) {

                r += "\nBase:\n";

                r2 += "\nBase:\n";
            }
        }

        return r + "\n" + r2;
    }
}

main_Square2.java

package main;

/**
 * Created by lucas on 22/08/16.
 */
public class Square2 {

    public int[] pecas = new int[]{0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1,
                                    0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1};

    public void u(int movimento){
        for (int i = 0; i < 12 - movimento; i++){

            moveTop();
        }
    }

    public void d(int move){
        for (int i = 0; i < 12 - move; i++){

            moverBase();
        }
    }

    public void s(){
        int[] x = pecas;

        int startOfPerms = 3;

        for (int i = startOfPerms; i < 6+startOfPerms; i++){

            int aux = x[i];

            x[i] = x[i+11];

            x[i+11] = aux;
        }
        pecas = x;
    }

    public boolean isBlocked(){
        //TODO functional and worker isTravado method here...
        return false;
    }

    private void moveTop() {
        int[] x = pecas;

        for (int i = 0; i < (x.length / 2) - 1; i++) {

            int aux = x[i];

            x[i] = x[i + 1];

            x[i + 1] = aux;
        }

        pecas = x;
    }

    private void moverBase(){
        int[] x = pecas;

        for (int i = 12; i < x.length; i++) {

            int aux = x[i];

            x[i] = x[i + 1];

            x[i + 1] = aux;
        }

        pecas = x;
    }

    @Override
    public String toString() {
        String r = "";

        for (int i = 0; i < pecas.length; i++){

            r += pecas[i] + "   ";

            if (i == (pecas.length / 2) - 1){

                r += "\n";
            }
        }
        return r;
    }
}

main_SquareScrambler.java

package main;

import java.util.ArrayList;
import java.util.Random;

/**
 * Created by lucas on 23/08/16.
 */
public class SquareScrambler {

    private Square square;
    private Random r = new Random();
    private ArrayList<int[]> pairs = new ArrayList<>();
    private int[] moves = new int[12];

    public SquareScrambler(Square square){
        this.square = square;
        for (int i = -5; i <= 6; i++){
            moves[i + 5] = i;
        }
    }

    public void teste1(){
        int u, d;

        do {
            u = moves[r.nextInt(moves.length)];
            d = moves[r.nextInt(moves.length)];
        } while (u == 0 && d == 0);

        square.u(u);
        square.d(d);

        if (square.isTravado()){
            square.u(u * -1);
            square.d(d * -1);
            teste1();
        } else {
            square.s();
            if (!square.isQuadrado()){

            } else {
                pairs.add(new int[]{u,d});
            }
        }
    }

    public void teste2(){

    }

    public String getSequence(){
        for (int i = 0; i < 10; i++){
            teste1();
        }

        String r = "";

        for (int[] x : pairs){
            r += "(" + x[0] + "," + x[1] + ")/";
        }

        return r;
    }
}

main_SquareShapeSolver.java

package main;

/**
 * Created by lucas on 23/08/16.
 */
public class SquareShapeSolver {
    //TODO something...
}

main_Type.java

package main;

/**
 * Created by lucas on 22/08/16.
 */
public enum Type {

    CANTO("canto", (byte)1), MEIO("meio", (byte)0);

    private String typeName;

    private byte nameAsByte;

    Type(String typeName, byte nameAsByte){
        this.typeName = typeName;

        this.nameAsByte = nameAsByte;
    }

    public String getTypeName() {
        return typeName;
    }

    public byte getNameAsByte() {
        return nameAsByte;
    }
}