飞行棋是一款经典的棋盘竞技游戏，玩家通过掷骰子驱动棋子在环形轨道上前进，以全部棋子抵达终点为胜。看似简单规则的背后，隐藏着丰富的概率计算与策略博弈空间：何时出动新棋子、是否冒险穿越敌阵、怎样选择最优移动目标，都需要综合评估风险与收益。本文将用 Java 完整实现飞行棋的核心逻辑，并重点讲解概率建模、路径规划与蒙特卡洛模拟在最优策略决策中的应用。
一、游戏规则与核心算法挑战
标准飞行棋棋盘由 52 个公共格子与 4 条各 6 格的终点通道组成。每位玩家拥有 4 枚棋子，起始于自家停机坪。核心规则如下：
- 掷出 6 点 可从停机坪出动一枚新棋子至起飞格，并获得额外一次掷骰机会
- 棋子按顺时针方向在环形公共轨道上移动，移动步数等于骰子点数
- 若棋子恰好落在敌方棋子所在格，敌方棋子被击落回停机坪（己方安全区与重叠格除外）
- 棋子绕行一周后进入自家终点通道，需恰好掷到剩余步数才能抵达终点
算法挑战：
- 概率建模：骰子结果的离散分布直接影响出动概率与期望步数
- 路径规划：环形轨道上的最短路径与多棋子协同移动策略
- 风险评估：穿越敌阵时的被击落概率量化
- 最优决策：多枚棋子可移动时，如何选择收益最大的走法
二、数据模型设计
首先建立棋盘、棋子与玩家的基础数据模型。
/**
* 棋子状态
*/
public class Piece {
public enum Status { HANGAR, BOARD, HOME } // 停机坪/轨道上/已到家
public int id; // 棋子编号 0-3
public int playerId; // 所属玩家 0-3
public Status status;
public int position; // 在轨道上的位置 (0-51)，仅在 BOARD 状态有效
public int homePosition; // 在终点通道中的位置 (0-5)，进入 HOME 后使用
public Piece(int id, int playerId) {
this.id = id;
this.playerId = playerId;
this.status = Status.HANGAR;
this.position = -1;
this.homePosition = -1;
}
}
/**
* 玩家
*/
public class Player {
public int id;
public String name;
public Piece[] pieces = new Piece[4];
public int startOffset; // 该玩家的起飞格在全局轨道上的偏移
public Player(int id, String name, int startOffset) {
this.id = id;
this.name = name;
this.startOffset = startOffset;
for (int i = 0; i < 4; i++) {
pieces[i] = new Piece(i, id);
}
}
/**
* 计算从当前位置移动到终点的剩余步数
*/
public int stepsToHome(Piece piece) {
if (piece.status == Piece.Status.HANGAR) return Integer.MAX_VALUE;
if (piece.status == Piece.Status.HOME) return 6 - piece.homePosition;
// 在公共轨道上，计算到进入终点通道的距离
int enterPos = (startOffset + 50) % 52; // 终点通道入口前一格
int dist = (enterPos - piece.position + 52) % 52;
return dist + 6; // +6 为终点通道长度
}
}
/**
* 棋盘
*/
public class Board {
public static final int TRACK_LENGTH = 52; // 公共轨道长度
public static final int HOME_LENGTH = 6; // 终点通道长度
public static final int PLAYER_COUNT = 4;
// 安全区位置：每个玩家的起飞格及重叠格
private final boolean[] safeZones = new boolean[TRACK_LENGTH];
public Board() {
// 四个玩家的起飞格为安全区
for (int i = 0; i < PLAYER_COUNT; i++) {
safeZones[i * 13] = true;
}
}
public boolean isSafe(int position) {
return safeZones[position % TRACK_LENGTH];
}
/**
* 获取指定位置上的所有棋子（用于碰撞检测）
*/
public List<Piece> getPiecesAt(int position, Player[] players) {
List<Piece> list = new ArrayList<>();
for (Player p : players) {
for (Piece piece : p.pieces) {
if (piece.status == Piece.Status.BOARD && piece.position == position) {
list.add(piece);
}
}
}
return list;
}
}
三、概率建模：骰子分布与期望计算
飞行棋中，骰子结果直接决定行动空间。我们建立概率模型来量化不同决策的长期收益。
/**
* 骰子概率模型
*/
public class DiceModel {
public static final int SIDES = 6;
/**
* 单骰各点数的概率质量函数 (公平骰子)
*/
public static double probability(int face) {
if (face < 1 || face > SIDES) return 0.0;
return 1.0 / SIDES;
}
/**
* 掷出 6 点的概率（出动新棋子的关键事件）
*/
public static double rollSixProbability() {
return 1.0 / SIDES;
}
/**
* 连续掷出 k 次 6 点的概率（理论上无上限，实际可截断）
*/
public static double consecutiveSixes(int k) {
return Math.pow(1.0 / SIDES, k);
}
/**
* 单轮行动中至少出动一枚新棋子的概率
* 即：至少掷出一个 6 点（考虑连续掷 6 的额外机会）
* P = 1/6 + (1/6)^2 + (1/6)^3 + ... = (1/6) / (1 - 1/6) = 1/5
*/
public static double launchProbabilityPerTurn() {
double p = 1.0 / SIDES;
return p / (1 - p); // 几何级数求和
}
/**
* 从轨道某位置恰好走到目标位置的掷骰概率
* 在 HOME 阶段需要精确步数
*/
public static double exactRollProbability(int needed) {
if (needed >= 1 && needed <= SIDES) return probability(needed);
return 0.0;
}
}
期望步数分析
当棋子在公共轨道上距离终点通道入口还有 d 步时，单轮期望前进步数为：
$$E[\text{步数}] = \sum_{i=1}^{6} i \cdot P(X=i) = 3.5$$
但由于存在被击落回停机坪的风险，有效期望步数需要结合存活概率修正。
四、路径规划与风险评估
在飞行棋中，最优路径不仅指最短几何路径，更意味着风险最小、收益最大的战略通道。
/**
* 风险评估与路径规划
*/
public class RiskEvaluator {
/**
* 评估移动指定棋子后的局面得分
* 得分越高，该走法越优
*/
public static double evaluateMove(Board board, Player[] players,
Player currentPlayer, Piece piece, int steps) {
double score = 0.0;
// 模拟移动后的位置
MoveResult result = simulateMove(board, players, currentPlayer, piece, steps);
Piece simulated = result.piece;
// 1. 抵达终点的奖励（最高优先级）
if (simulated.status == Piece.Status.HOME && simulated.homePosition == Board.HOME_LENGTH) {
score += 10000.0;
}
// 2. 进入终点通道的奖励
else if (simulated.status == Piece.Status.HOME) {
score += 500.0 * simulated.homePosition;
}
// 3. 击落敌方棋子的奖励
if (result.capturedPieces > 0) {
score += 800.0 * result.capturedPieces;
}
// 4. 前进距离的线性奖励
if (simulated.status == Piece.Status.BOARD) {
score += 10.0 * steps;
// 5. 位置安全性评估
if (board.isSafe(simulated.position)) {
score += 200.0; // 安全区加分
} else {
// 6. 被击落风险惩罚
double risk = calculateCaptureRisk(board, players, currentPlayer, simulated.position);
score -= 600.0 * risk;
}
// 7. 靠近终点通道入口的加速奖励
int stepsToHome = currentPlayer.stepsToHome(simulated);
if (stepsToHome < 20) {
score += 50.0 * (20 - stepsToHome);
}
}
// 8. 从停机坪出动的奖励（前期优先出兵）
if (piece.status == Piece.Status.HANGAR && steps == 6) {
long boardCount = Arrays.stream(currentPlayer.pieces)
.filter(p -> p.status == Piece.Status.BOARD || p.status == Piece.Status.HOME)
.count();
if (boardCount < 2) {
score += 300.0; // 前期优先多出兵
}
}
return score;
}
/**
* 计算某位置被敌方下一回合击落的概率
*/
public static double calculateCaptureRisk(Board board, Player[] players,
Player currentPlayer, int position) {
double risk = 0.0;
for (Player enemy : players) {
if (enemy.id == currentPlayer.id) continue;
for (Piece p : enemy.pieces) {
if (p.status != Piece.Status.BOARD) continue;
// 敌方需要掷出恰好 (position - enemyPos + 52) % 52 才能到达
int needed = (position - p.position + Board.TRACK_LENGTH) % Board.TRACK_LENGTH;
if (needed >= 1 && needed <= 6) {
risk += DiceModel.probability(needed);
}
}
}
return Math.min(risk, 1.0);
}
/**
* 模拟移动结果
*/
private static MoveResult simulateMove(Board board, Player[] players,
Player player, Piece piece, int steps) {
// 深拷贝棋子进行模拟
Piece sim = new Piece(piece.id, piece.playerId);
sim.status = piece.status;
sim.position = piece.position;
sim.homePosition = piece.homePosition;
int captured = 0;
if (sim.status == Piece.Status.HANGAR && steps == 6) {
// 出动至起飞格
sim.status = Piece.Status.BOARD;
sim.position = player.startOffset;
} else if (sim.status == Piece.Status.BOARD) {
int newPos = (sim.position + steps) % Board.TRACK_LENGTH;
int enterPos = (player.startOffset + 50) % Board.TRACK_LENGTH;
// 判断是否进入终点通道
int distToEnter = (enterPos - sim.position + Board.TRACK_LENGTH) % Board.TRACK_LENGTH;
if (steps > distToEnter) {
int homeSteps = steps - distToEnter - 1;
if (homeSteps < Board.HOME_LENGTH) {
sim.status = Piece.Status.HOME;
sim.homePosition = homeSteps;
}
// 超出则不可移动（规则上通常不允许或停在入口前）
} else {
sim.position = newPos;
// 碰撞检测与击落
if (!board.isSafe(newPos)) {
for (Player other : players) {
if (other.id == player.id) continue;
for (Piece enemy : other.pieces) {
if (enemy.status == Piece.Status.BOARD && enemy.position == newPos) {
captured++;
}
}
}
}
}
} else if (sim.status == Piece.Status.HOME) {
int newHome = sim.homePosition + steps;
if (newHome <= Board.HOME_LENGTH) {
sim.homePosition = newHome;
if (newHome == Board.HOME_LENGTH) {
sim.status = Piece.Status.HOME; // 已到家
}
}
}
return new MoveResult(sim, captured);
}
static class MoveResult {
Piece piece;
int capturedPieces;
MoveResult(Piece p, int c) { this.piece = p; this.capturedPieces = c; }
}
}
五、蒙特卡洛模拟：AI 最优决策
当存在多枚可移动的棋子时，我们需要选择最优走法。蒙特卡洛模拟通过大量随机推演来评估每个候选走法的期望胜率。
/**
* 蒙特卡洛模拟决策引擎
*/
public class MonteCarloDecisionEngine {
private static final int SIMULATION_COUNT = 5000; // 模拟次数
private final Random random = new Random();
/**
* 为当前玩家选择最优走法
* @return 最优决策 [pieceId, expectedWinRate]
*/
public double[] selectBestMove(Board board, Player[] players, int currentPlayerId, int diceRoll) {
Player current = players[currentPlayerId];
List<Integer> movablePieces = getMovablePieces(current, diceRoll);
if (movablePieces.isEmpty()) return new double[]{-1, 0.0};
if (movablePieces.size() == 1) return new double[]{movablePieces.get(0), 1.0};
double bestWinRate = -1.0;
int bestPiece = -1;
for (int pieceId : movablePieces) {
double winRate = simulateWinRate(board, players, currentPlayerId, pieceId, diceRoll);
if (winRate > bestWinRate) {
bestWinRate = winRate;
bestPiece = pieceId;
}
}
return new double[]{bestPiece, bestWinRate};
}
/**
* 对指定走法进行蒙特卡洛模拟，估算胜率
*/
private double simulateWinRate(Board board, Player[] players, int currentPlayerId,
int pieceId, int diceRoll) {
int wins = 0;
for (int i = 0; i < SIMULATION_COUNT; i++) {
// 深拷贝当前局面
GameState state = cloneState(board, players);
// 执行候选走法
executeMove(state, currentPlayerId, pieceId, diceRoll);
// 快速模拟后续对局至结束
if (fastSimulate(state, currentPlayerId)) {
wins++;
}
}
return (double) wins / SIMULATION_COUNT;
}
/**
* 快速模拟：使用贪心策略快速推演至游戏结束
*/
private boolean fastSimulate(GameState state, int targetPlayerId) {
int turns = 0;
int current = 0;
while (turns < 500) { // 防止无限循环
// 检查胜利条件
if (hasWon(state.players[targetPlayerId])) return true;
if ( Arrays.stream(state.players).anyMatch(p -> p.id != targetPlayerId && hasWon(p)) ) {
return false;
}
// 掷骰子
int roll = random.nextInt(6) + 1;
Player player = state.players[current];
List<Integer> movable = getMovablePieces(player, roll);
if (!movable.isEmpty()) {
// 快速模拟中使用启发式评估选择走法（非完整 MCTS）
int best = movable.get(0);
double bestScore = -Double.MAX_VALUE;
for (int pid : movable) {
double s = RiskEvaluator.evaluateMove(
state.board, state.players, player, player.pieces[pid], roll);
if (s > bestScore) {
bestScore = s;
best = pid;
}
}
executeMove(state, current, best, roll);
}
// 6 点获得额外回合（简化处理：最多连续 3 次）
if (roll != 6 || turns % 3 == 2) {
current = (current + 1) % Board.PLAYER_COUNT;
}
turns++;
}
// 超时判定：比较各玩家已到家棋子数
return countHomePieces(state.players[targetPlayerId]) >=
Arrays.stream(state.players).filter(p -> p.id != targetPlayerId)
.mapToInt(this::countHomePieces).max().orElse(0);
}
private List<Integer> getMovablePieces(Player player, int diceRoll) {
List<Integer> list = new ArrayList<>();
for (int i = 0; i < 4; i++) {
Piece p = player.pieces[i];
if (p.status == Piece.Status.HOME && p.homePosition == Board.HOME_LENGTH) continue;
if (p.status == Piece.Status.HANGAR && diceRoll == 6) list.add(i);
else if (p.status == Piece.Status.BOARD) list.add(i);
else if (p.status == Piece.Status.HOME && p.homePosition + diceRoll <= Board.HOME_LENGTH) {
list.add(i);
}
}
return list;
}
private boolean hasWon(Player player) {
return Arrays.stream(player.pieces)
.allMatch(p -> p.status == Piece.Status.HOME && p.homePosition == Board.HOME_LENGTH);
}
private int countHomePieces(Player player) {
return (int) Arrays.stream(player.pieces)
.filter(p -> p.status == Piece.Status.HOME && p.homePosition == Board.HOME_LENGTH)
.count();
}
// 深拷贝与执行走法的辅助方法（省略具体实现）
private GameState cloneState(Board board, Player[] players) { /* ... */ return null; }
private void executeMove(GameState state, int playerId, int pieceId, int steps) { /* ... */ }
static class GameState { Board board; Player[] players; }
}
六、游戏主循环与控制台演示
/**
* 飞行棋游戏引擎
*/
public class AeroplaneChess {
private final Board board = new Board();
private final Player[] players = new Player[Board.PLAYER_COUNT];
private final Random dice = new Random();
private final MonteCarloDecisionEngine ai = new MonteCarloDecisionEngine();
public AeroplaneChess() {
String[] names = {"红方", "黄方", "蓝方", "绿方"};
for (int i = 0; i < Board.PLAYER_COUNT; i++) {
players[i] = new Player(i, names[i], i * 13);
}
}
public void play() {
int current = 0;
int round = 0;
while (round < 1000) {
Player player = players[current];
System.out.println("\n===== " + player.name + " 的回合 =====");
int consecutiveSixes = 0;
while (consecutiveSixes < 3) {
int roll = dice.nextInt(6) + 1;
System.out.println("掷出: " + roll);
double[] decision = ai.selectBestMove(board, players, current, roll);
if (decision[0] < 0) {
System.out.println("无可用走法");
break;
}
int pieceId = (int) decision[0];
System.out.printf("选择棋子 %d，预估胜率: %.2f%%\n", pieceId, decision[1] * 100);
movePiece(player.pieces[pieceId], roll);
printBoard();
if (hasWon(player)) {
System.out.println("🎉 " + player.name + " 获胜！");
return;
}
if (roll == 6) {
consecutiveSixes++;
System.out.println("获得额外回合！");
} else {
break;
}
}
current = (current + 1) % Board.PLAYER_COUNT;
round++;
}
}
private void movePiece(Piece piece, int steps) {
// 整合前面的移动逻辑（含碰撞检测）
}
private void printBoard() {
// 简化控制台输出
}
private boolean hasWon(Player player) {
return Arrays.stream(player.pieces)
.allMatch(p -> p.status == Piece.Status.HOME && p.homePosition == Board.HOME_LENGTH);
}
public static void main(String[] args) {
new AeroplaneChess().play();
}
}
七、复杂度分析
| 模块 | 时间复杂度 | 空间复杂度 | 说明 |
|---|---|---|---|
| 单次局面评估 | O(P) | O(1) | P 为场上棋子总数（≤16） |
| 蒙特卡洛单次模拟 | O(T · P) | O(P) | T 为模拟轮数上限 |
| 完整决策（5000 次模拟） | O(S · T · P) | O(P) | S=5000 为模拟次数 |
| 棋盘上棋子移动 | O(1) | O(1) | 哈希定位 |
优化方向：
- 置换表（Transposition Table）：缓存已评估的局面，避免重复计算
- 剪枝策略：当某走法胜率显著低于当前最优时提前终止模拟
- 并行化：利用多线程并行执行蒙特卡洛模拟，决策速度可提升 N 倍（N 为 CPU 核心数）
八、总结
飞行棋的魅力在于概率不确定性与策略确定性的交织。本文从工程实现角度，建立了完整的概率模型、风险评估体系与蒙特卡洛决策引擎：
- 概率建模量化了出动期望与精确抵达概率，为战略节奏提供数学依据
- 路径规划综合距离、安全性与击落收益，构建了多维度局面评估函数
- 蒙特卡洛模拟通过大量随机推演收敛出最优走法，使 AI 具备接近人类高手的决策水平
读者可以在此基础上继续扩展：引入多线程并行模拟、设计更精细的评估函数权重、或加入在线学习机制让 AI 从历史对局中持续进化。
