引言
三阶魔方（Rubik’s Cube）是匈牙利建筑学教授鲁比克于1974年发明的经典三维组合 puzzle。一个标准三阶魔方拥有约4.3×10¹⁹种不同的状态，其还原问题本质上是在一个巨大的状态空间图中寻找从任意状态到目标状态的最短路径。本文将用Java实现三阶魔方的核心数据结构、群论表示方法，以及经典的Kociemba两阶段搜索算法，带你深入理解这个NP-hard问题的高效求解策略。
一、三阶魔方的数学建模
1.1 魔方的面与块表示
三阶魔方由26个可见小块组成：6个中心块（固定不动）、8个角块（每个角块有3种颜色）、12个棱块（每个棱块有2种颜色）。中心块的颜色决定了该面的最终颜色。
我们可以用以下方式编码魔方状态：
– 角块位置：8个角块按固定顺序编号，每个角块有3种朝向
– 棱块位置：12个棱块按固定顺序编号，每个棱块有2种朝向
1.2 群论视角
魔方的所有合法转动构成了一个置换群。六个基本转动（U、D、L、R、F、B）是群的生成元。每个转动都是对20个可动块（8角+12棱）的置换操作。
Kociemba算法的核心洞察是：将魔方群分解为两个子群的乘积。第一阶段将任意状态转换到”G1群”（只包含
二、Java核心数据结构
import java.util.*;
/**
* 三阶魔方状态表示类
* 使用角块和棱块的排列与朝向来编码状态
*/
public class CubeState {
// 角块排列：8个角块的位置（0-7）
// 角块朝向：每个角块的朝向（0-2），模3
byte[] cp; // corner permutation
byte[] co; // corner orientation
// 棱块排列：12个棱块的位置（0-11）
// 棱块朝向：每个棱块的朝向（0-1），模2
byte[] ep; // edge permutation
byte[] eo; // edge orientation
// 预计算的转动表（通过静态初始化生成）
static final CubeMove[] MOVES = CubeMove.values();
public CubeState() {
// 初始状态：所有块都在正确位置，朝向为0
this.cp = new byte[]{0,1,2,3,4,5,6,7};
this.co = new byte[]{0,0,0,0,0,0,0,0};
this.ep = new byte[]{0,1,2,3,4,5,6,7,8,9,10,11};
this.eo = new byte[]{0,0,0,0,0,0,0,0,0,0,0,0};
}
/**
* 深拷贝构造函数
*/
public CubeState(CubeState other) {
this.cp = other.cp.clone();
this.co = other.co.clone();
this.ep = other.ep.clone();
this.eo = other.eo.clone();
}
/**
* 应用一个基本转动，返回新状态
* @param move 转动操作
* @return 转动后的新状态
*/
public CubeState applyMove(CubeMove move) {
CubeState next = new CubeState(this);
move.apply(next);
return next;
}
/**
* 计算角块排列的序号（用于模式数据库索引）
* 使用Lehmer编码将排列映射为0..8!-1的整数
*/
public int cornerPermutationIndex() {
int index = 0;
for (int i = 0; i < 8; i++) {
int count = 0;
for (int j = i + 1; j < 8; j++) {
if (cp[j] < cp[i]) count++;
}
index = index * (8 - i) + count;
}
return index;
}
/**
* 计算棱块排列的序号（用于模式数据库索引）
*/
public int edgePermutationIndex() {
int index = 0;
for (int i = 0; i < 12; i++) {
int count = 0;
for (int j = i + 1; j < 12; j++) {
if (ep[j] < ep[i]) count++;
}
index = index * (12 - i) + count;
}
return index;
}
@Override
public boolean equals(Object o) {
if (this == o) return true;
if (!(o instanceof CubeState)) return false;
CubeState other = (CubeState) o;
return Arrays.equals(cp, other.cp) && Arrays.equals(co, other.co)
&& Arrays.equals(ep, other.ep) && Arrays.equals(eo, other.eo);
}
@Override
public int hashCode() {
return Arrays.hashCode(cp) * 31 + Arrays.hashCode(co) * 17
+ Arrays.hashCode(ep) * 13 + Arrays.hashCode(eo) * 7;
}
}
/**
* 魔方基本转动枚举
* 每个枚举值包含对应的状态变换逻辑
*/
enum CubeMove {
U, U2, U3, // U面转动 90/180/270度
D, D2, D3,
L, L2, L3,
R, R2, R3,
F, F2, F3,
B, B2, B3;
// 角块转动映射表：每个操作对应的角块位置循环和朝向变化
private static final byte[][] CORNER_PERM_TABLE = initializeCornerPermTable();
private static final byte[][] CORNER_ORI_TABLE = initializeCornerOriTable();
private static final byte[][] EDGE_PERM_TABLE = initializeEdgePermTable();
private static final byte[][] EDGE_ORI_TABLE = initializeEdgeOriTable();
/**
* 初始化角块排列映射表
* U操作循环角块0-1-2-3，其余不变
* 具体映射通过标准魔方标记法确定
*/
private static byte[][] initializeCornerPermTable() {
byte[][] table = new byte[18][8];
// U: (0,1,2,3) -> (1,2,3,0)
table[0] = new byte[]{1,2,3,0,4,5,6,7};
// U2: (0,1,2,3) -> (2,3,0,1)
table[1] = new byte[]{2,3,0,1,4,5,6,7};
// U3: (0,1,2,3) -> (3,0,1,2)
table[2] = new byte[]{3,0,1,2,4,5,6,7};
// D: (4,5,6,7) -> (5,6,7,4)
table[3] = new byte[]{0,1,2,3,5,6,7,4};
table[4] = new byte[]{0,1,2,3,6,7,4,5};
table[5] = new byte[]{0,1,2,3,7,4,5,6};
// L: (0,3,4,7) -> (3,4,7,0)
table[6] = new byte[]{3,1,2,7,0,5,6,4};
table[7] = new byte[]{7,1,2,4,3,5,6,0};
table[8] = new byte[]{4,1,2,0,7,5,6,3};
// R: (1,2,5,6) -> (2,5,6,1)
table[9] = new byte[]{0,2,6,3,4,1,5,7};
table[10] = new byte[]{0,6,5,3,4,2,1,7};
table[11] = new byte[]{0,5,1,3,4,6,2,7};
// F: (0,1,5,4) -> (1,5,4,0)
table[12] = new byte[]{1,5,2,3,0,4,6,7};
table[13] = new byte[]{5,4,2,3,1,0,6,7};
table[14] = new byte[]{4,0,2,3,5,1,6,7};
// B: (2,3,7,6) -> (3,7,6,2)
table[15] = new byte[]{0,1,7,6,4,5,2,3};
table[16] = new byte[]{0,1,6,2,4,5,7,3};
table[17] = new byte[]{0,1,3,7,4,5,6,2};
return table;
}
private static byte[][] initializeCornerOriTable() {
byte[][] table = new byte[18][8];
// U和D操作不改变角块朝向
for (int i = 0; i < 6; i++) Arrays.fill(table[i], (byte)0);
// L操作：(0,3,4,7)的朝向变化
table[6] = new byte[]{2,0,0,1,2,0,0,1}; // +2 for entering, +1 for leaving
table[7] = new byte[]{0,0,0,0,0,0,0,0};
table[8] = new byte[]{1,0,0,2,1,0,0,2};
// R操作
table[9] = new byte[]{0,1,2,0,0,1,2,0};
table[10] = new byte[]{0,0,0,0,0,0,0,0};
table[11] = new byte[]{0,2,1,0,0,2,1,0};
// F操作
table[12] = new byte[]{1,2,0,0,1,2,0,0};
table[13] = new byte[]{0,0,0,0,0,0,0,0};
table[14] = new byte[]{2,1,0,0,2,1,0,0};
// B操作
table[15] = new byte[]{0,0,1,2,0,0,1,2};
table[16] = new byte[]{0,0,0,0,0,0,0,0};
table[17] = new byte[]{0,0,2,1,0,0,2,1};
return table;
}
private static byte[][] initializeEdgePermTable() {
byte[][] table = new byte[18][12];
// U: (0,1,2,3) edges
table[0] = new byte[]{1,2,3,0,4,5,6,7,8,9,10,11};
table[1] = new byte[]{2,3,0,1,4,5,6,7,8,9,10,11};
table[2] = new byte[]{3,0,1,2,4,5,6,7,8,9,10,11};
// D: (8,9,10,11)
table[3] = new byte[]{0,1,2,3,4,5,6,7,9,10,11,8};
table[4] = new byte[]{0,1,2,3,4,5,6,7,10,11,8,9};
table[5] = new byte[]{0,1,2,3,4,5,6,7,11,8,9,10};
// L: (0,4,8,7)
table[6] = new byte[]{4,1,2,3,8,5,6,0,7,9,10,11};
table[7] = new byte[]{8,1,2,3,7,5,6,4,0,9,10,11};
table[8] = new byte[]{7,1,2,3,0,5,6,8,4,9,10,11};
// R: (2,6,10,5)
table[9] = new byte[]{0,1,6,3,4,2,10,7,8,9,5,11};
table[10] = new byte[]{0,1,10,3,4,6,5,7,8,9,2,11};
table[11] = new byte[]{0,1,5,3,4,10,2,7,8,9,6,11};
// F: (3,7,11,6)
table[12] = new byte[]{0,1,2,7,4,5,3,11,8,9,10,6};
table[13] = new byte[]{0,1,2,11,4,5,7,6,8,9,10,3};
table[14] = new byte[]{0,1,2,6,4,5,11,3,8,9,10,7};
// B: (1,5,9,4)
table[15] = new byte[]{0,5,2,3,9,1,6,7,8,4,10,11};
table[16] = new byte[]{0,1,2,3,5,9,6,7,8,1,10,11}; // simplified
table[17] = new byte[]{0,9,2,3,1,5,6,7,8,5,10,11}; // simplified
return table;
}
private static byte[][] initializeEdgeOriTable() {
byte[][] table = new byte[18][12];
// U和D不改变棱块朝向
for (int i = 0; i < 6; i++) Arrays.fill(table[i], (byte)0);
// L, R, F, B改变特定棱块的朝向
table[6] = new byte[]{1,0,0,0,1,0,0,1,1,0,0,0};
table[7] = new byte[]{0,0,0,0,0,0,0,0,0,0,0,0};
table[8] = new byte[]{1,0,0,0,1,0,0,1,1,0,0,0};
table[9] = new byte[]{0,0,1,0,0,1,1,0,0,0,1,0};
table[10] = new byte[]{0,0,0,0,0,0,0,0,0,0,0,0};
table[11] = new byte[]{0,0,1,0,0,1,1,0,0,0,1,0};
table[12] = new byte[]{0,0,0,1,0,0,1,1,0,0,0,1};
table[13] = new byte[]{0,0,0,0,0,0,0,0,0,0,0,0};
table[14] = new byte[]{0,0,0,1,0,0,1,1,0,0,0,1};
table[15] = new byte[]{0,1,0,0,1,1,0,0,0,1,0,0};
table[16] = new byte[]{0,0,0,0,0,0,0,0,0,0,0,0};
table[17] = new byte[]{0,1,0,0,1,1,0,0,0,1,0,0};
return table;
}
/**
* 在当前状态上应用本转动操作
*/
public void apply(CubeState state) {
int idx = this.ordinal();
byte[] newCp = new byte[8];
byte[] newCo = new byte[8];
byte[] newEp = new byte[12];
byte[] newEo = new byte[12];
// 应用角块排列变换
for (int i = 0; i < 8; i++) {
newCp[i] = state.cp[CORNER_PERM_TABLE[idx][i]];
newCo[i] = (byte)((state.co[CORNER_PERM_TABLE[idx][i]] + CORNER_ORI_TABLE[idx][i]) % 3);
}
// 应用棱块排列变换
for (int i = 0; i < 12; i++) {
newEp[i] = state.ep[EDGE_PERM_TABLE[idx][i]];
newEo[i] = (byte)((state.eo[EDGE_PERM_TABLE[idx][i]] + EDGE_ORI_TABLE[idx][i]) % 2);
}
state.cp = newCp;
state.co = newCo;
state.ep = newEp;
state.eo = newEo;
}
}
三、Kociemba两阶段搜索算法
3.1 算法核心思想
Kociemba算法（又称二阶段算法）将还原过程分为两个阶段：
第一阶段：使用
第二阶段：仅使用
3.2 模式数据库（Pattern Database）
为了高效搜索，我们预计算两个模式数据库：
1. 第一阶段数据库：记录每个状态到达G1群的最少步数（上限为7步）
2. 第二阶段数据库：记录在G1群内到达初始状态的最少步数
由于完整数据库太大，实践中使用启发式估计。
/**
* Kociemba两阶段搜索求解器
*/
public class KociembaSolver {
// 第一阶段的18个允许操作（不含L2,R2,F2,B2）
private static final CubeMove[] PHASE1_MOVES = {
CubeMove.U, CubeMove.U2, CubeMove.U3,
CubeMove.D, CubeMove.D2, CubeMove.D3,
CubeMove.L, CubeMove.L3,
CubeMove.R, CubeMove.R3,
CubeMove.F, CubeMove.F3,
CubeMove.B, CubeMove.B3
};
// 第二阶段的10个允许操作
private static final CubeMove[] PHASE2_MOVES = {
CubeMove.U, CubeMove.U2, CubeMove.U3,
CubeMove.D, CubeMove.D2, CubeMove.D3,
CubeMove.L2, CubeMove.R2, CubeMove.F2, CubeMove.B2
};
private final int maxDepthPhase1 = 7;
private final int maxDepthPhase2 = 13;
/**
* 检查状态是否属于G1群（第一阶段目标）
* G1群条件：co全为0，eo全为0，UD层棱块与E层棱块不混合
*/
public boolean isInG1(CubeState state) {
// 角块朝向全为0
for (byte o : state.co) if (o != 0) return false;
// 棱块朝向全为0
for (byte o : state.eo) if (o != 0) return false;
// U/D层棱块(0,1,2,3,8,9,10,11)与E层棱块(4,5,6,7)不混合
// 简化检查：UD层棱块位置都在0-3或8-11范围内
for (int i = 0; i < 4; i++) {
int pos = state.ep[i];
if (pos < 0 || pos > 3) return false; // 应该是UD层棱块
}
for (int i = 8; i < 12; i++) {
int pos = state.ep[i];
if (pos < 8 || pos > 11) return false;
}
return true;
}
/**
* 第一阶段的启发式估计：计算到G1群的估计距离
* 基于角块朝向错误数、棱块朝向错误数和棱块层混合数
*/
public int phase1Heuristic(CubeState state) {
int h = 0;
// 角块朝向错误（每个错误至少1步）
for (byte o : state.co) if (o != 0) h++;
// 棱块朝向错误
for (byte o : state.eo) if (o != 0) h++;
// 棱块层混合（每个错误至少2步）
// 简化：返回一个下界估计
return Math.max(h / 4, 1); // 粗略估计
}
/**
* 第二阶段启发式：在G1群内到目标状态的估计距离
*/
public int phase2Heuristic(CubeState state) {
// 在G1群内，只需考虑角块排列和棱块排列
int cornerDist = 0;
for (int i = 0; i < 8; i++) {
if (state.cp[i] != i) cornerDist++;
}
int edgeDist = 0;
for (int i = 0; i < 12; i++) {
if (state.ep[i] != i) edgeDist++;
}
return Math.max(cornerDist / 2, edgeDist / 2);
}
/**
* 使用IDA*搜索第一阶段路径
* @param start 起始状态
* @param path 输出路径
* @return 是否找到路径
*/
public boolean searchPhase1(CubeState start, List<CubeMove> path) {
int bound = phase1Heuristic(start);
while (bound <= maxDepthPhase1) {
int t = idaStarPhase1(start, 0, bound, path, null);
if (t == -1) return true; // 找到解
if (t == Integer.MAX_VALUE) return false; // 无解
bound = t;
}
return false;
}
private int idaStarPhase1(CubeState state, int g, int bound,
List<CubeMove> path, CubeMove lastMove) {
int f = g + phase1Heuristic(state);
if (f > bound) return f;
if (isInG1(state)) return -1; // 到达G1群
int min = Integer.MAX_VALUE;
for (CubeMove move : PHASE1_MOVES) {
// 剪枝：避免连续转动同一面或相反面
if (lastMove != null && !isValidSequence(lastMove, move)) continue;
path.add(move);
CubeState next = state.applyMove(move);
int t = idaStarPhase1(next, g + 1, bound, path, move);
if (t == -1) return -1;
if (t < min) min = t;
path.remove(path.size() - 1);
}
return min;
}
/**
* 检查两个操作是否构成有效序列（避免冗余）
* 规则：不连续转动同一面，不连续转动相对面
*/
private boolean isValidSequence(CubeMove last, CubeMove current) {
int lastFace = last.ordinal() / 3;
int currFace = current.ordinal() / 3;
// 0=U,1=D,2=L,3=R,4=F,5=B
if (lastFace == currFace) return false; // 同一面
if ((lastFace == 0 && currFace == 1) || (lastFace == 1 && currFace == 0)) return false;
if ((lastFace == 2 && currFace == 3) || (lastFace == 3 && currFace == 2)) return false;
if ((lastFace == 4 && currFace == 5) || (lastFace == 5 && currFace == 4)) return false;
return true;
}
/**
* 完整的两阶段求解
*/
public List<CubeMove> solve(CubeState scrambled) {
List<CubeMove> phase1Path = new ArrayList<>();
if (!searchPhase1(scrambled, phase1Path)) {
throw new RuntimeException("Phase 1 search failed");
}
// 应用第一阶段结果得到G1群状态
CubeState g1State = new CubeState(scrambled);
for (CubeMove m : phase1Path) {
g1State = g1State.applyMove(m);
}
// 第二阶段搜索（在G1群内）
List<CubeMove> phase2Path = new ArrayList<>();
// 简化：使用BFS或IDA*在G1群内搜索
if (!searchPhase2(g1State, phase2Path)) {
throw new RuntimeException("Phase 2 search failed");
}
List<CubeMove> solution = new ArrayList<>(phase1Path);
solution.addAll(phase2Path);
return solution;
}
/**
* 第二阶段搜索（简化版BFS）
*/
public boolean searchPhase2(CubeState start, List<CubeMove> path) {
Queue<CubeState> queue = new LinkedList<>();
Map<CubeState, CubeState> parent = new HashMap<>();
Map<CubeState, CubeMove> moveUsed = new HashMap<>();
queue.add(start);
parent.put(start, null);
CubeState goal = new CubeState();
int steps = 0;
while (!queue.isEmpty() && steps < maxDepthPhase2) {
int size = queue.size();
for (int i = 0; i < size; i++) {
CubeState cur = queue.poll();
if (cur.equals(goal)) {
// 回溯路径
CubeState s = cur;
List<CubeMove> revPath = new ArrayList<>();
while (parent.get(s) != null) {
revPath.add(moveUsed.get(s));
s = parent.get(s);
}
Collections.reverse(revPath);
path.addAll(revPath);
return true;
}
for (CubeMove move : PHASE2_MOVES) {
CubeState next = cur.applyMove(move);
if (!parent.containsKey(next)) {
parent.put(next, cur);
moveUsed.put(next, move);
queue.add(next);
}
}
}
steps++;
}
return false;
}
}
四、完整可运行示例
/**
* 三阶魔方求解器主程序
*/
public class RubiksCubeSolver {
public static void main(String[] args) {
// 创建初始状态
CubeState solved = new CubeState();
System.out.println("初始状态（已还原）:");
printState(solved);
// 打乱魔方：应用一系列随机转动
CubeMove[] scramble = {
CubeMove.R, CubeMove.U, CubeMove.R3, CubeMove.F2,
CubeMove.L, CubeMove.D2, CubeMove.B, CubeMove.U3
};
CubeState scrambled = new CubeState(solved);
System.out.println("\n打乱序列:");
for (CubeMove m : scramble) {
System.out.print(m + " ");
scrambled = scrambled.applyMove(m);
}
System.out.println();
System.out.println("\n打乱后状态:");
printState(scrambled);
// 使用Kociemba算法求解
KociembaSolver solver = new KociembaSolver();
long startTime = System.currentTimeMillis();
try {
List<CubeMove> solution = solver.solve(scrambled);
long endTime = System.currentTimeMillis();
System.out.println("\n求解成功！步数: " + solution.size());
System.out.println("求解时间: " + (endTime - startTime) + "ms");
System.out.println("还原步骤:");
for (CubeMove m : solution) {
System.out.print(m + " ");
}
System.out.println();
// 验证解的正确性
CubeState verify = new CubeState(scrambled);
for (CubeMove m : solution) {
verify = verify.applyMove(m);
}
if (verify.equals(solved)) {
System.out.println("\n验证通过：魔方已成功还原！");
}
} catch (RuntimeException e) {
System.out.println("求解失败: " + e.getMessage());
System.out.println("提示：完整的Kociemba实现需要更大的模式数据库和更精细的启发式函数");
}
}
private static void printState(CubeState state) {
System.out.println(" 角块排列: " + Arrays.toString(state.cp));
System.out.println(" 角块朝向: " + Arrays.toString(state.co));
System.out.println(" 棱块排列: " + Arrays.toString(state.ep));
System.out.println(" 棱块朝向: " + Arrays.toString(state.eo));
}
}
五、算法复杂度分析
| 指标 | 复杂度 | 说明 |
|---|---|---|
| 状态空间 | 4.3×10¹⁹ | 三阶魔方的所有合法状态 |
| 上帝数 | 20 | 任意状态最多20步还原（2010年证明） |
| 第一阶段搜索 | O(18^d) | d为第一阶段深度，通常d≤7 |
| 第二阶段搜索 | O(10^d) | d为第二阶段深度，通常d≤13 |
| 实际求解时间 | <100ms | 现代计算机配合模式数据库可在毫秒级求解 |
六、总结与扩展
本文实现了三阶魔方的核心数据结构和Kociemba两阶段搜索算法框架。完整的高效实现还需要：
1. 更大的模式数据库：预计算更多状态的精确距离
2. 对称性约简：利用魔方的48种对称性减少搜索空间
3. 多线程搜索：并行搜索不同的分支
4. 更精细的启发式：结合多个模式数据库取最大值
读者可以通过构建完整的模式数据库，将本框架扩展为一个真正能在毫秒级求解任意打乱状态的三阶魔方求解器。
