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

  1. Public Class SudokuClass
  2. Public Property Rank As Integer '数独的阶数
  3. Get
  4. Return GridRank
  5. End Get
  6. Set(ByVal value As Integer)
  7. GridRank = value
  8. SubGridRank = CType(Math.Sqrt(value), Integer)
  9. SubGridIncludeRank = SubGridRank
  10. DataMaxValue = value
  11. DataMinValue = 1
  12. End Set
  13. End Property
  14. Public Property GridData As Integer?(,) '数独的数据
  15. Private GridRank As Integer '数独的阶数
  16. Private SubGridRank As Integer '子数独(宫)的阶数
  17. Private SubGridIncludeRank As Integer '数独包含子数独(宫)的阶数
  18. Private DataMaxValue As Integer '数独可填最大数值
  19. Private DataMinValue As Integer '数独可填最小数值
  20. ''' <summary>
  21. ''' 实例化一个指定阶数的数独类
  22. ''' </summary>
  23. ''' <param name="nRank">指定的阶数</param>
  24. Public Sub New(nRank As Integer)
  25. Me.Rank = nRank
  26. End Sub
  27. End Class

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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。

  1. '查找当前空白格(最佳格)
  2. Private Function getStartPoint(ByRef data As Integer?(,)) As Point
  3. Dim gPoint As Point
  4. Dim tempValue As Integer
  5. Dim maxValue As Integer
  6. '查找限制最多的空白格
  7. For i = 0 To GridRank - 1
  8. For j = 0 To GridRank - 1
  9. If data(i, j) = 0 Then
  10. tempValue = 0
  11. For k = 0 To GridRank - 1
  12. If data(i, k) > 0 Then tempValue += 1
  13. If data(k, j) > 0 Then tempValue += 1
  14. If data((i \ SubGridIncludeRank) * SubGridIncludeRank + k \ SubGridIncludeRank, (j \ SubGridIncludeRank) * SubGridIncludeRank + (k Mod SubGridIncludeRank)) > 0 Then tempValue += 1
  15. Next
  16. If tempValue > maxValue Then
  17. maxValue = tempValue
  18. gPoint.X = i
  19. gPoint.Y = j
  20. End If
  21. End If
  22. Next
  23. Next
  24. If maxValue > 0 Then
  25. Return gPoint
  26. Else
  27. gPoint.X = -1
  28. gPoint.Y = -1
  29. Return gPoint
  30. End If
  31. End Function

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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。

  1. '判断同行同列同宫是否已经存在
  2. Private Function GetExisting(ByRef data As Integer?(,), ByVal gX As Integer, ByVal gY As Integer, ByVal gValue As Integer) As Boolean
  3. For k = 0 To GridRank - 1
  4. 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
  5. Return True
  6. End If
  7. Next
  8. Return False
  9. End Function

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2）回溯法求解

简单的递归调用即可。

  1. '递归求解数独
  2. Private Function GetValue(ByVal gData As Integer?(,)) As List(Of Integer?(,))
  3. Dim ResultList As New List(Of Integer?(,))
  4. Dim i, j As Integer
  5. Dim tempPoint As Point = getStartPoint(gData)
  6. i = tempPoint.X : j = tempPoint.Y
  7. If i >= 0 AndAlso j >= 0 Then
  8. For value = DataMinValue To DataMaxValue
  9. If GetExisting(gData, i, j, value) = False Then
  10. gData(i, j) = value
  11. GetValue(gData)
  12. gData(i, j) = 0
  13. End If
  14. Next
  15. Else
  16. '新增一个结果
  17. ResultList.Add(gData)
  18. End If
  19. Return ResultList
  20. End Function

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3）其他

判断一个序列（数组）中是否有重复数字

  1. ''' <summary>
  2. ''' 判断一个序列是否有重复数字
  3. ''' </summary>
  4. ''' <param name="Numbers"></param>
  5. ''' <returns></returns>
  6. Private Function IsDuplicate(ByVal Numbers() As Integer) As Boolean
  7. Array.Sort(Numbers)
  8. If Numbers.Length > 1 Then
  9. For i = 0 To Numbers.Length - 2
  10. If Numbers(i) = Numbers(i + 1) Then Return True
  11. Next
  12. End If
  13. Return False
  14. End Function

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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

  1. Public Function IsWin(ByVal Numbers As Integer?(,)) As Boolean
  2. For i = 0 To GridRank - 1
  3. For j = 0 To GridRank - 1
  4. If Not Numbers(i, j).HasValue Then Return False '出现空格
  5. Next
  6. Next
  7. Dim TempInt As New List(Of Integer)
  8. '判断行重复
  9. For i = 0 To GridRank - 1
  10. TempInt.Clear()
  11. For j = 0 To GridRank - 1
  12. TempInt.Add(CInt(Numbers(i, j)))
  13. Next
  14. If IsDuplicate(TempInt.ToArray) Then Return False
  15. Next
  16. '判断列重复
  17. For j = 0 To GridRank - 1
  18. TempInt.Clear()
  19. For i = 0 To GridRank - 1
  20. TempInt.Add(CInt(Numbers(i, j)))
  21. Next
  22. If IsDuplicate(TempInt.ToArray) Then Return False
  23. Next
  24. '判断宫格重复
  25. For i = 0 To GridRank - 1 Step SubGridRank
  26. For j = 0 To GridRank - 1 Step SubGridRank
  27. TempInt.Clear()
  28. For i2 = 0 To SubGridRank - 1
  29. For j2 = 0 To SubGridRank - 1
  30. TempInt.Add(CInt(Numbers(i + i2, j + j2)))
  31. Next
  32. Next
  33. If IsDuplicate(TempInt.ToArray) Then Return False
  34. Next
  35. Next
  36. Return True
  37. End Function

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判断当前数独是否符合规则，检查手机游戏拍卖每一个非空白格的数字的同行、同列、同宫是否重复。

  1. Public Function IsImpossible(ByVal Numbers As Integer?(,)) As Boolean
  2. Dim temp As Integer?
  3. For i = 0 To GridRank - 1
  4. For j = 0 To GridRank - 1
  5. If Not Numbers(i, j) = 0 Then
  6. temp = Numbers(i, j)
  7. Numbers(i, j) = 0
  8. If GetExisting(Numbers, i, j, CInt(temp)) Then Numbers(i, j) = temp : Return True
  9. Numbers(i, j) = temp
  10. End If
  11. Next
  12. Next
  13. Return False
  14. End Function

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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）

  1. Public Class SudokuClass
  2. Public Property Rank As Integer '数独的阶数
  3. Get
  4. Return GridRank
  5. End Get
  6. Set(ByVal value As Integer)
  7. GridRank = value
  8. SubGridRank = CType(Math.Sqrt(value), Integer)
  9. SubGridIncludeRank = SubGridRank
  10. DataMaxValue = value
  11. DataMinValue = 1
  12. End Set
  13. End Property
  14. Public Property GridData As Integer?(,) '数独的数据
  15. Private GridRank As Integer '数独的阶数
  16. Private SubGridRank As Integer '子数独(宫)的阶数
  17. Private SubGridIncludeRank As Integer '数独包含子数独(宫)的阶数
  18. Private DataMaxValue As Integer '数独可填最大数值
  19. Private DataMinValue As Integer '数独可填最小数值
  20. ''' <summary>
  21. ''' 实例化一个指定阶数的数独类
  22. ''' </summary>
  23. ''' <param name="nRank">指定的阶数</param>
  24. Public Sub New(nRank As Integer)
  25. Me.Rank = nRank
  26. End Sub
  27. Public Function GenerateInitialNumbers() As Integer?(,)
  28. ReDim GridData(GridRank - 1, GridRank - 1)
  29. For i = 0 To GridRank - 1
  30. For j = 0 To GridRank - 1
  31. GridData(i, j) = 0 '暂无初始数字生成规则，请从数独文件导入
  32. Next
  33. Next
  34. Return GridData '返回一个空白数独
  35. End Function
  36. Public Function IsImpossible(ByVal Numbers As Integer?(,)) As Boolean
  37. Dim temp As Integer?
  38. For i = 0 To GridRank - 1
  39. For j = 0 To GridRank - 1
  40. If Not Numbers(i, j) = 0 Then
  41. temp = Numbers(i, j)
  42. Numbers(i, j) = 0
  43. If GetExisting(Numbers, i, j, CInt(temp)) Then Numbers(i, j) = temp : Return True
  44. Numbers(i, j) = temp
  45. End If
  46. Next
  47. Next
  48. Return False
  49. End Function
  50. Public Function IsWin(ByVal Numbers As Integer?(,)) As Boolean
  51. For i = 0 To GridRank - 1
  52. For j = 0 To GridRank - 1
  53. If Not Numbers(i, j).HasValue Then Return False '出现空格
  54. Next
  55. Next
  56. Dim TempInt As New List(Of Integer)
  57. '判断行重复
  58. For i = 0 To GridRank - 1
  59. TempInt.Clear()
  60. For j = 0 To GridRank - 1
  61. TempInt.Add(CInt(Numbers(i, j)))
  62. Next
  63. If IsDuplicate(TempInt.ToArray) Then Return False
  64. Next
  65. '判断列重复
  66. For j = 0 To GridRank - 1
  67. TempInt.Clear()
  68. For i = 0 To GridRank - 1
  69. TempInt.Add(CInt(Numbers(i, j)))
  70. Next
  71. If IsDuplicate(TempInt.ToArray) Then Return False
  72. Next
  73. '判断宫格重复
  74. For i = 0 To GridRank - 1 Step SubGridRank
  75. For j = 0 To GridRank - 1 Step SubGridRank
  76. TempInt.Clear()
  77. For i2 = 0 To SubGridRank - 1
  78. For j2 = 0 To SubGridRank - 1
  79. TempInt.Add(CInt(Numbers(i + i2, j + j2)))
  80. Next
  81. Next
  82. If IsDuplicate(TempInt.ToArray) Then Return False
  83. Next
  84. Next
  85. Return True
  86. End Function
  87. ''' <summary>
  88. ''' 判断一个序列是否有重复数字
  89. ''' </summary>
  90. ''' <param name="Numbers"></param>
  91. ''' <returns></returns>
  92. Private Function IsDuplicate(ByVal Numbers() As Integer) As Boolean
  93. Array.Sort(Numbers)
  94. If Numbers.Length > 1 Then
  95. For i = 0 To Numbers.Length - 2
  96. If Numbers(i) = Numbers(i + 1) Then Return True
  97. Next
  98. End If
  99. Return False
  100. End Function
  101. ''' <summary>
  102. ''' 返回指定位置的所有可填数字的序列
  103. ''' </summary>
  104. ''' <param name="Numbers">原数组</param>
  105. ''' <param name="gX">指定的位置的X值，从0开始</param>
  106. ''' <param name="gY">指定的位置的Y值，从0开始</param>
  107. ''' <returns></returns>
  108. Public Function GetEnabledNum(ByVal Numbers As Integer?(,), gX As Integer, gY As Integer) As Integer()
  109. Dim NumList As New List(Of Integer)
  110. For i = DataMinValue To DataMaxValue
  111. If GetExisting(Numbers, gX, gY, i) = False Then NumList.Add(i)
  112. Next
  113. Return NumList.ToArray
  114. End Function
  115. '递归求解数独
  116. Private Function GetValue(ByVal gData As Integer?(,)) As List(Of Integer?(,))
  117. Dim ResultList As New List(Of Integer?(,))
  118. Dim i, j As Integer
  119. Dim tempPoint As Point = getStartPoint(gData)
  120. i = tempPoint.X : j = tempPoint.Y
  121. If i >= 0 AndAlso j >= 0 Then
  122. For value = DataMinValue To DataMaxValue
  123. If GetExisting(gData, i, j, value) = False Then
  124. gData(i, j) = value
  125. GetValue(gData)
  126. gData(i, j) = 0
  127. End If
  128. Next
  129. Else
  130. '新增一个结果
  131. ResultList.Add(gData)
  132. End If
  133. Return ResultList
  134. End Function
  135. '查找当前空白格(最佳格)
  136. Private Function getStartPoint(ByRef data As Integer?(,)) As Point
  137. Dim gPoint As Point
  138. Dim tempValue As Integer
  139. Dim maxValue As Integer
  140. '查找限制最多的空白格
  141. For i = 0 To GridRank - 1
  142. For j = 0 To GridRank - 1
  143. If data(i, j) = 0 Then
  144. tempValue = 0
  145. For k = 0 To GridRank - 1
  146. If data(i, k) > 0 Then tempValue += 1
  147. If data(k, j) > 0 Then tempValue += 1
  148. If data((i \ SubGridIncludeRank) * SubGridIncludeRank + k \ SubGridIncludeRank, (j \ SubGridIncludeRank) * SubGridIncludeRank + (k Mod SubGridIncludeRank)) > 0 Then tempValue += 1
  149. Next
  150. If tempValue > maxValue Then
  151. maxValue = tempValue
  152. gPoint.X = i
  153. gPoint.Y = j
  154. End If
  155. End If
  156. Next
  157. Next
  158. If maxValue > 0 Then
  159. Return gPoint
  160. Else
  161. gPoint.X = -1
  162. gPoint.Y = -1
  163. Return gPoint
  164. End If
  165. End Function
  166. '判断同行同列同宫是否已经存在
  167. Private Function GetExisting(ByRef data As Integer?(,), ByVal gX As Integer, ByVal gY As Integer, ByVal gValue As Integer) As Boolean
  168. For k = 0 To GridRank - 1
  169. 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
  170. Return True
  171. End If
  172. Next
  173. Return False
  174. End Function
  175. End Class

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  1. using Microsoft.VisualBasic;
  2. using System;
  3. using System.Collections;
  4. using System.Collections.Generic;
  5. using System.Data;
  6. using System.Diagnostics;
  7. public class SudokuClass
  8. {
  9. public int Rank {
  10. //数独的阶数
  11. get { return GridRank; }
  12. set {
  13. GridRank = value;
  14. SubGridRank = Convert.ToInt32(Math.Sqrt(value));
  15. SubGridIncludeRank = SubGridRank;
  16. DataMaxValue = value;
  17. DataMinValue = 1;
  18. }
  19. }
  20. public int?[,] GridData { get; set; }
  21. //数独的数据
  22. //数独的阶数
  23. private int GridRank;
  24. //子数独(宫)的阶数
  25. private int SubGridRank;
  26. //数独包含子数独(宫)的阶数
  27. private int SubGridIncludeRank;
  28. //数独可填最大数值
  29. private int DataMaxValue;
  30. //数独可填最小数值
  31. private int DataMinValue;
  32. /// <summary>
  33. /// 实例化一个指定阶数的数独类
  34. /// </summary>
  35. /// <param name="nRank">指定的阶数</param>
  36. public SudokuClass(int nRank)
  37. {
  38. this.Rank = nRank;
  39. }
  40. public int?[,] GenerateInitialNumbers()
  41. {
  42. GridData = new Nullable<int>[GridRank, GridRank];
  43. for (i = 0; i <= GridRank - 1; i++) {
  44. for (j = 0; j <= GridRank - 1; j++) {
  45. GridData[i, j] = 0;
  46. //暂无初始数字生成规则，请从数独文件导入
  47. }
  48. }
  49. return GridData;
  50. //返回一个空白数独
  51. }
  52. public bool IsImpossible(int?[,] Numbers)
  53. {
  54. int? temp = default(int?);
  55. for (i = 0; i <= GridRank - 1; i++) {
  56. for (j = 0; j <= GridRank - 1; j++) {
  57. if (!(Numbers[i, j] == 0)) {
  58. temp = Numbers[i, j];
  59. Numbers[i, j] = 0;
  60. if (GetExisting(ref Numbers, i, j, Convert.ToInt32(temp))){Numbers[i, j] = temp;return true;}
  61. Numbers[i, j] = temp;
  62. }
  63. }
  64. }
  65. return false;
  66. }
  67. public bool IsWin(int?[,] Numbers)
  68. {
  69. for (i = 0; i <= GridRank - 1; i++) {
  70. for (j = 0; j <= GridRank - 1; j++) {
  71. if (!Numbers[i, j].HasValue)
  72. return false;
  73. //出现空格
  74. }
  75. }
  76. List<int> TempInt = new List<int>();
  77. //判断行重复
  78. for (i = 0; i <= GridRank - 1; i++) {
  79. TempInt.Clear();
  80. for (j = 0; j <= GridRank - 1; j++) {
  81. TempInt.Add(Convert.ToInt32(Numbers[i, j]));
  82. }
  83. if (IsDuplicate(TempInt.ToArray()))
  84. return false;
  85. }
  86. //判断列重复
  87. for (j = 0; j <= GridRank - 1; j++) {
  88. TempInt.Clear();
  89. for (i = 0; i <= GridRank - 1; i++) {
  90. TempInt.Add(Convert.ToInt32(Numbers[i, j]));
  91. }
  92. if (IsDuplicate(TempInt.ToArray()))
  93. return false;
  94. }
  95. //判断宫格重复
  96. for (i = 0; i <= GridRank - 1; i += SubGridRank) {
  97. for (j = 0; j <= GridRank - 1; j += SubGridRank) {
  98. TempInt.Clear();
  99. for (i2 = 0; i2 <= SubGridRank - 1; i2++) {
  100. for (j2 = 0; j2 <= SubGridRank - 1; j2++) {
  101. TempInt.Add(Convert.ToInt32(Numbers[i + i2, j + j2]));
  102. }
  103. }
  104. if (IsDuplicate(TempInt.ToArray()))
  105. return false;
  106. }
  107. }
  108. return true;
  109. }
  110. /// <summary>
  111. /// 判断一个序列是否有重复数字
  112. /// </summary>
  113. /// <param name="Numbers"></param>
  114. /// <returns></returns>
  115. private bool IsDuplicate(int[] Numbers)
  116. {
  117. Array.Sort(Numbers);
  118. if (Numbers.Length > 1) {
  119. for (i = 0; i <= Numbers.Length - 2; i++) {
  120. if (Numbers[i] == Numbers[i + 1])
  121. return true;
  122. }
  123. }
  124. return false;
  125. }
  126. /// <summary>
  127. /// 返回指定位置的所有可填数字的序列
  128. /// </summary>
  129. /// <param name="Numbers">原数组</param>
  130. /// <param name="gX">指定的位置的X值，从0开始</param>
  131. /// <param name="gY">指定的位置的Y值，从0开始</param>
  132. /// <returns></returns>
  133. public int[] GetEnabledNum(int?[,] Numbers, int gX, int gY)
  134. {
  135. List<int> NumList = new List<int>();
  136. for (i = DataMinValue; i <= DataMaxValue; i++) {
  137. if (GetExisting(ref Numbers, gX, gY, i) == false)
  138. NumList.Add(i);
  139. }
  140. return NumList.ToArray();
  141. }
  142. //递归求解数独
  143. private List<int?[,]> GetValue(int?[,] gData)
  144. {
  145. List<int?[,]> ResultList = new List<int?[,]>();
  146. int i = 0;
  147. int j = 0;
  148. Point tempPoint = getStartPoint(ref gData);
  149. i = tempPoint.X;
  150. j = tempPoint.Y;
  151. if (i >= 0 && j >= 0) {
  152. for (value = DataMinValue; value <= DataMaxValue; value++) {
  153. if (GetExisting(ref gData, i, j, value) == false) {
  154. gData[i, j] = value;
  155. GetValue(gData);
  156. gData[i, j] = 0;
  157. }
  158. }
  159. } else {
  160. //新增一个结果
  161. ResultList.Add(gData);
  162. }
  163. return ResultList;
  164. }
  165. //查找当前空白格(最佳格)
  166. private Point getStartPoint(ref int?[,] data)
  167. {
  168. Point gPoint = default(Point);
  169. int tempValue = 0;
  170. int maxValue = 0;
  171. //查找限制最多的空白格
  172. for (i = 0; i <= GridRank - 1; i++) {
  173. for (j = 0; j <= GridRank - 1; j++) {
  174. if (data[i, j] == 0) {
  175. tempValue = 0;
  176. for (k = 0; k <= GridRank - 1; k++) {
  177. if (data[i, k] > 0)
  178. tempValue += 1;
  179. if (data[k, j] > 0)
  180. tempValue += 1;
  181. if (data[(i / SubGridIncludeRank) * SubGridIncludeRank + k / SubGridIncludeRank, (j / SubGridIncludeRank) * SubGridIncludeRank + (k % SubGridIncludeRank)] > 0)
  182. tempValue += 1;
  183. }
  184. if (tempValue > maxValue) {
  185. maxValue = tempValue;
  186. gPoint.X = i;
  187. gPoint.Y = j;
  188. }
  189. }
  190. }
  191. }
  192. if (maxValue > 0) {
  193. return gPoint;
  194. } else {
  195. gPoint.X = -1;
  196. gPoint.Y = -1;
  197. return gPoint;
  198. }
  199. }
  200. //判断同行同列同宫是否已经存在
  201. private bool GetExisting(ref int?[,] data, int gX, int gY, int gValue)
  202. {
  203. for (k = 0; k <= GridRank - 1; k++) {
  204. if (data[gX, k] == gValue || data[k, gY] == gValue || data[(gX / SubGridIncludeRank) * SubGridIncludeRank + k / SubGridIncludeRank, (gY / SubGridIncludeRank) * SubGridIncludeRank + (k % SubGridIncludeRank)] == gValue) {
  205. return true;
  206. }
  207. }
  208. return false;
  209. }
  210. }

复制代码

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