数独游戏求解：解法适用于任意阶数的数独
0.数独简介
数独（すうどく，Sūdoku）是一种运用纸、笔进行演算的逻辑游戏。以九阶数独为例，玩家需要根据9×9盘面上的已知数字，推理出所有剩余空格的数字，并满足每一行、每一列、每一个粗线宫内的数字均含1-9，不重复。
1）4阶（可填数字范围1~4，宫格2阶）
2）9阶（可填数字范围1~9，宫格3阶）
3）16阶（可填数字范围1~16，宫格4阶）
*见附录
1.数独的表示
对于N阶数独可以用一个N*N的二维数组表示
1）数独阶数GridRank=N
2）宫格阶数SubGridRank=Sqrt(N)
3）数独包含宫的阶数SubGridIncludeRank=Sqrt(N)
4）可填最大数字MaxValue=N
5）可填最小数字MinValue=1
Public Class SudokuClass Public Property Rank As Integer '数独的阶数 Get Return GridRank End Get Set(ByVal value As Integer) GridRank = value SubGridRank = CType(Math.Sqrt(value), Integer) SubGridIncludeRank = SubGridRank DataMaxValue = value DataMinValue = 1 End Set End Property Public Property GridData As Integer?(,) '数独的数据 Private GridRank As Integer '数独的阶数 Private SubGridRank As Integer '子数独(宫)的阶数 Private SubGridIncludeRank As Integer '数独包含子数独(宫)的阶数 Private DataMaxValue As Integer '数独可填最大数值 Private DataMinValue As Integer '数独可填最小数值 ''' <summary> ''' 实例化一个指定阶数的数独类 ''' </summary> ''' <param name="nRank">指定的阶数</param> Public Sub New(nRank As Integer) Me.Rank = nRank End Sub End Class
6）任意阶数独的表示（N≠K2,K>1,K∈Z）
任意阶数独仅有一个宫，所以数独阶数和宫阶数相等。以7阶为例，设置GridRank=7,SubGridRank=7,SubGridIncludeRank=1即可。
2.数独的求解
采用回溯法，直接寻找到一个空白格，尝试填入所有可能的数字，继续填下一个空白格，直至填满或者不能继续填入为止（不符合规则）。
1）寻找空白格
空白格的选择至关重要，当一个格子的限制越多,也就是可填入数字越少时，优先选择该空白格。这样做可以大大降低递归填格子的次数。
'查找当前空白格(最佳格) Private Function getStartPoint(ByRef data As Integer?(,)) As Point Dim gPoint As Point Dim tempValue As Integer Dim maxValue As Integer '查找限制最多的空白格 For i = 0 To GridRank - 1 For j = 0 To GridRank - 1 If data(i, j) = 0 Then tempValue = 0 For k = 0 To GridRank - 1 If data(i, k) > 0 Then tempValue += 1 If data(k, j) > 0 Then tempValue += 1 If data((i \ SubGridIncludeRank) * SubGridIncludeRank + k \ SubGridIncludeRank, (j \ SubGridIncludeRank) * SubGridIncludeRank + (k Mod SubGridIncludeRank)) > 0 Then tempValue += 1 Next If tempValue > maxValue Then maxValue = tempValue gPoint.X = i gPoint.Y = j End If End If Next Next If maxValue > 0 Then Return gPoint Else gPoint.X = -1 gPoint.Y = -1 Return gPoint End If End Function
当确定空白格后就需要向其中填入数字了，一一筛选出符合规则的数字。规则就是格子当前行，当前列，当前宫不能有相同数值出现。
'判断同行同列同宫是否已经存在 Private Function GetExisting(ByRef data As Integer?(,), ByVal gX As Integer, ByVal gY As Integer, ByVal gValue As Integer) As Boolean For k = 0 To GridRank - 1 If data(gX, k) = gValue OrElse data(k, gY) = gValue OrElse data((gX \ SubGridIncludeRank) * SubGridIncludeRank + k \ SubGridIncludeRank, (gY \ SubGridIncludeRank) * SubGridIncludeRank + (k Mod SubGridIncludeRank)) = gValue Then Return True End If Next Return False End Function
2）回溯法求解
简单的递归调用即可。
'递归求解数独 Private Function GetValue(ByVal gData As Integer?(,)) As List(Of Integer?(,)) Dim ResultList As New List(Of Integer?(,)) Dim i, j As Integer Dim tempPoint As Point = getStartPoint(gData) i = tempPoint.X : j = tempPoint.Y If i >= 0 AndAlso j >= 0 Then For value = DataMinValue To DataMaxValue If GetExisting(gData, i, j, value) = False Then gData(i, j) = value GetValue(gData) gData(i, j) = 0 End If Next Else '新增一个结果 ResultList.Add(gData) End If Return ResultList End Function
3）其他
判断一个序列（数组）中是否有重复数字
''' <summary> ''' 判断一个序列是否有重复数字 ''' </summary> ''' <param name="Numbers"></param> ''' <returns></returns> Private Function IsDuplicate(ByVal Numbers() As Integer) As Boolean Array.Sort(Numbers) If Numbers.Length > 1 Then For i = 0 To Numbers.Length - 2 If Numbers(i) = Numbers(i + 1) Then Return True Next End If Return False End Function
判断数独是否填写完毕，无空白格且所有格子填入的数字符合规则
Public Function IsWin(ByVal Numbers As Integer?(,)) As Boolean For i = 0 To GridRank - 1 For j = 0 To GridRank - 1 If Not Numbers(i, j).HasValue Then Return False '出现空格 Next Next Dim TempInt As New List(Of Integer) '判断行重复 For i = 0 To GridRank - 1 TempInt.Clear() For j = 0 To GridRank - 1 TempInt.Add(CInt(Numbers(i, j))) Next If IsDuplicate(TempInt.ToArray) Then Return False Next '判断列重复 For j = 0 To GridRank - 1 TempInt.Clear() For i = 0 To GridRank - 1 TempInt.Add(CInt(Numbers(i, j))) Next If IsDuplicate(TempInt.ToArray) Then Return False Next '判断宫格重复 For i = 0 To GridRank - 1 Step SubGridRank For j = 0 To GridRank - 1 Step SubGridRank TempInt.Clear() For i2 = 0 To SubGridRank - 1 For j2 = 0 To SubGridRank - 1 TempInt.Add(CInt(Numbers(i + i2, j + j2))) Next Next If IsDuplicate(TempInt.ToArray) Then Return False Next Next Return True End Function
判断当前数独是否符合规则，检查每一个非空白格的数字的同行、同列、同宫是否重复。
Public Function IsImpossible(ByVal Numbers As Integer?(,)) As Boolean Dim temp As Integer? For i = 0 To GridRank - 1 For j = 0 To GridRank - 1 If Not Numbers(i, j) = 0 Then temp = Numbers(i, j) Numbers(i, j) = 0 If GetExisting(Numbers, i, j, CInt(temp)) Then Numbers(i, j) = temp : Return True Numbers(i, j) = temp End If Next Next Return False End Function
附录
1）16阶（可填数字范围1~16，宫格4阶）
2）VB.NET类 & C#类(在线工具转换，仅供参考）
Public Class SudokuClass Public Property Rank As Integer '数独的阶数 Get Return GridRank End Get Set(ByVal value As Integer) GridRank = value SubGridRank = CType(Math.Sqrt(value), Integer) SubGridIncludeRank = SubGridRank DataMaxValue = value DataMinValue = 1 End Set End Property Public Property GridData As Integer?(,) '数独的数据 Private GridRank As Integer '数独的阶数 Private SubGridRank As Integer '子数独(宫)的阶数 Private SubGridIncludeRank As Integer '数独包含子数独(宫)的阶数 Private DataMaxValue As Integer '数独可填最大数值 Private DataMinValue As Integer '数独可填最小数值 ''' <summary> ''' 实例化一个指定阶数的数独类 ''' </summary> ''' <param name="nRank">指定的阶数</param> Public Sub New(nRank As Integer) Me.Rank = nRank End Sub Public Function GenerateInitialNumbers() As Integer?(,) ReDim GridData(GridRank - 1, GridRank - 1) For i = 0 To GridRank - 1 For j = 0 To GridRank - 1 GridData(i, j) = 0 '暂无初始数字生成规则，请从数独文件导入 Next Next Return GridData '返回一个空白数独 End Function Public Function IsImpossible(ByVal Numbers As Integer?(,)) As Boolean Dim temp As Integer? For i = 0 To GridRank - 1 For j = 0 To GridRank - 1 If Not Numbers(i, j) = 0 Then temp = Numbers(i, j) Numbers(i, j) = 0 If GetExisting(Numbers, i, j, CInt(temp)) Then Numbers(i, j) = temp : Return True Numbers(i, j) = temp End If Next Next Return False End Function Public Function IsWin(ByVal Numbers As Integer?(,)) As Boolean For i = 0 To GridRank - 1 For j = 0 To GridRank - 1 If Not Numbers(i, j).HasValue Then Return False '出现空格 Next Next Dim TempInt As New List(Of Integer) '判断行重复 For i = 0 To GridRank - 1 TempInt.Clear() For j = 0 To GridRank - 1 TempInt.Add(CInt(Numbers(i, j))) Next If IsDuplicate(TempInt.ToArray) Then Return False Next '判断列重复 For j = 0 To GridRank - 1 TempInt.Clear() For i = 0 To GridRank - 1 TempInt.Add(CInt(Numbers(i, j))) Next If IsDuplicate(TempInt.ToArray) Then Return False Next '判断宫格重复 For i = 0 To GridRank - 1 Step SubGridRank For j = 0 To GridRank - 1 Step SubGridRank TempInt.Clear() For i2 = 0 To SubGridRank - 1 For j2 = 0 To SubGridRank - 1 TempInt.Add(CInt(Numbers(i + i2, j + j2))) Next Next If IsDuplicate(TempInt.ToArray) Then Return False Next Next Return True End Function ''' <summary> ''' 判断一个序列是否有重复数字 ''' </summary> ''' <param name="Numbers"></param> ''' <returns></returns> Private Function IsDuplicate(ByVal Numbers() As Integer) As Boolean Array.Sort(Numbers) If Numbers.Length > 1 Then For i = 0 To Numbers.Length - 2 If Numbers(i) = Numbers(i + 1) Then Return True Next End If Return False End Function ''' <summary> ''' 返回指定位置的所有可填数字的序列 ''' </summary> ''' <param name="Numbers">原数组</param> ''' <param name="gX">指定的位置的X值，从0开始</param> ''' <param name="gY">指定的位置的Y值，从0开始</param> ''' <returns></returns> Public Function GetEnabledNum(ByVal Numbers As Integer?(,), gX As Integer, gY As Integer) As Integer() Dim NumList As New List(Of Integer) For i = DataMinValue To DataMaxValue If GetExisting(Numbers, gX, gY, i) = False Then NumList.Add(i) Next Return NumList.ToArray End Function '递归求解数独 Private Function GetValue(ByVal gData As Integer?(,)) As List(Of Integer?(,)) Dim ResultList As New List(Of Integer?(,)) Dim i, j As Integer Dim tempPoint As Point = getStartPoint(gData) i = tempPoint.X : j = tempPoint.Y If i >= 0 AndAlso j >= 0 Then For value = DataMinValue To DataMaxValue If GetExisting(gData, i, j, value) = False Then gData(i, j) = value GetValue(gData) gData(i, j) = 0 End If Next Else '新增一个结果 ResultList.Add(gData) End If Return ResultList End Function '查找当前空白格(最佳格) Private Function getStartPoint(ByRef data As Integer?(,)) As Point Dim gPoint As Point Dim tempValue As Integer Dim maxValue As Integer '查找限制最多的空白格 For i = 0 To GridRank - 1 For j = 0 To GridRank - 1 If data(i, j) = 0 Then tempValue = 0 For k = 0 To GridRank - 1 If data(i, k) > 0 Then tempValue += 1 If data(k, j) > 0 Then tempValue += 1 If data((i \ SubGridIncludeRank) * SubGridIncludeRank + k \ SubGridIncludeRank, (j \ SubGridIncludeRank) * SubGridIncludeRank + (k Mod SubGridIncludeRank)) > 0 Then tempValue += 1 Next If tempValue > maxValue Then maxValue = tempValue gPoint.X = i gPoint.Y = j End If End If Next Next If maxValue > 0 Then Return gPoint Else gPoint.X = -1 gPoint.Y = -1 Return gPoint End If End Function '判断同行同列同宫是否已经存在 Private Function GetExisting(ByRef data As Integer?(,), ByVal gX As Integer, ByVal gY As Integer, ByVal gValue As Integer) As Boolean For k = 0 To GridRank - 1 If data(gX, k) = gValue OrElse data(k, gY) = gValue OrElse data((gX \ SubGridIncludeRank) * SubGridIncludeRank + k \ SubGridIncludeRank, (gY \ SubGridIncludeRank) * SubGridIncludeRank + (k Mod SubGridIncludeRank)) = gValue Then Return True End If Next Return False End Function End Class
using Microsoft.VisualBasic; using System; using System.Collections; using System.Collections.Generic; using System.Data; using System.Diagnostics; public class SudokuClass { public int Rank { //数独的阶数 get { return GridRank; } set { GridRank = value; SubGridRank = Convert.ToInt32(Math.Sqrt(value)); SubGridIncludeRank = SubGridRank; DataMaxValue = value; DataMinValue = 1; } } public int?[,] GridData { get; set; } //数独的数据 //数独的阶数 private int GridRank; //子数独(宫)的阶数 private int SubGridRank; //数独包含子数独(宫)的阶数 private int SubGridIncludeRank; //数独可填最大数值 private int DataMaxValue; //数独可填最小数值 private int DataMinValue; /// <summary> /// 实例化一个指定阶数的数独类 /// </summary> /// <param name="nRank">指定的阶数</param> public SudokuClass(int nRank) { this.Rank = nRank; } public int?[,] GenerateInitialNumbers() { GridData = new Nullable<int>[GridRank, GridRank]; for (i = 0; i <= GridRank - 1; i++) { for (j = 0; j <= GridRank - 1; j++) { GridData[i, j] = 0; //暂无初始数字生成规则，请从数独文件导入 } } return GridData; //返回一个空白数独 } public bool IsImpossible(int?[,] Numbers) { int? temp = default(int?); for (i = 0; i <= GridRank - 1; i++) { for (j = 0; j <= GridRank - 1; j++) { if (!(Numbers[i, j] == 0)) { temp = Numbers[i, j]; Numbers[i, j] = 0; if (GetExisting(ref Numbers, i, j, Convert.ToInt32(temp))){Numbers[i, j] = temp;return true;} Numbers[i, j] = temp; } } } return false; } public bool IsWin(int?[,] Numbers) { for (i = 0; i <= GridRank - 1; i++) { for (j = 0; j <= GridRank - 1; j++) { if (!Numbers[i, j].HasValue) return false; //出现空格 } } List<int> TempInt = new List<int>(); //判断行重复 for (i = 0; i <= GridRank - 1; i++) { TempInt.Clear(); for (j = 0; j <= GridRank - 1; j++) { TempInt.Add(Convert.ToInt32(Numbers[i, j])); } if (IsDuplicate(TempInt.ToArray())) return false; } //判断列重复 for (j = 0; j <= GridRank - 1; j++) { TempInt.Clear(); for (i = 0; i <= GridRank - 1; i++) { TempInt.Add(Convert.ToInt32(Numbers[i, j])); } if (IsDuplicate(TempInt.ToArray())) return false; } //判断宫格重复 for (i = 0; i <= GridRank - 1; i += SubGridRank) { for (j = 0; j <= GridRank - 1; j += SubGridRank) { TempInt.Clear(); for (i2 = 0; i2 <= SubGridRank - 1; i2++) { for (j2 = 0; j2 <= SubGridRank - 1; j2++) { TempInt.Add(Convert.ToInt32(Numbers[i + i2, j + j2])); } } if (IsDuplicate(TempInt.ToArray())) return false; } } return true; } /// <summary> /// 判断一个序列是否有重复数字 /// </summary> /// <param name="Numbers"></param> /// <returns></returns> private bool IsDuplicate(int[] Numbers) { Array.Sort(Numbers); if (Numbers.Length > 1) { for (i = 0; i <= Numbers.Length - 2; i++) { if (Numbers[i] == Numbers[i + 1]) return true; } } return false; } /// <summary> /// 返回指定位置的所有可填数字的序列 /// </summary> /// <param name="Numbers">原数组</param> /// <param name="gX">指定的位置的X值，从0开始</param> /// <param name="gY">指定的位置的Y值，从0开始</param> /// <returns></returns> public int[] GetEnabledNum(int?[,] Numbers, int gX, int gY) { List<int> NumList = new List<int>(); for (i = DataMinValue; i <= DataMaxValue; i++) { if (GetExisting(ref Numbers, gX, gY, i) == false) NumList.Add(i); } return NumList.ToArray(); } //递归求解数独 private List<int?[,]> GetValue(int?[,] gData) { List<int?[,]> ResultList = new List<int?[,]>(); int i = 0; int j = 0; Point tempPoint = getStartPoint(ref gData); i = tempPoint.X; j = tempPoint.Y; if (i >= 0 && j >= 0) { for (value = DataMinValue; value <= DataMaxValue; value++) { if (GetExisting(ref gData, i, j, value) == false) { gData[i, j] = value; GetValue(gData); gData[i, j] = 0; } } } else { //新增一个结果 ResultList.Add(gData); } return ResultList; } //查找当前空白格(最佳格) private Point getStartPoint(ref int?[,] data) { Point gPoint = default(Point); int tempValue = 0; int maxValue = 0; //查找限制最多的空白格 for (i = 0; i <= GridRank - 1; i++) { for (j = 0; j <= GridRank - 1; j++) { if (data[i, j] == 0) { tempValue = 0; for (k = 0; k <= GridRank - 1; k++) { if (data[i, k] > 0) tempValue += 1; if (data[k, j] > 0) tempValue += 1; if (data[(i / SubGridIncludeRank) * SubGridIncludeRank + k / SubGridIncludeRank, (j / SubGridIncludeRank) * SubGridIncludeRank + (k % SubGridIncludeRank)] > 0) tempValue += 1; } if (tempValue > maxValue) { maxValue = tempValue; gPoint.X = i; gPoint.Y = j; } } } } if (maxValue > 0) { return gPoint; } else { gPoint.X = -1; gPoint.Y = -1; return gPoint; } } //判断同行同列同宫是否已经存在 private bool GetExisting(ref int?[,] data, int gX, int gY, int gValue) { for (k = 0; k <= GridRank - 1; k++) { if (data[gX, k] == gValue || data[k, gY] == gValue || data[(gX / SubGridIncludeRank) * SubGridIncludeRank + k / SubGridIncludeRank, (gY / SubGridIncludeRank) * SubGridIncludeRank + (k % SubGridIncludeRank)] == gValue) { return true; } } return false; } }