2017上海汽车零部件展:求教24点游戏的算法（最好用C++)

24点游戏是一个大众化的益智游戏.任意给四张扑克牌(不包括大小王),只能够用加,减,乘,除以及适当的括号连接这四张牌,无论顺序,使计算结果为24,或者宣布根本就是无解的.需要注意的是,每张牌必须运算,并且只能运算一次,J,Q,K可设置为11,12,13.
本程序目的就是算出一组牌的所有解(不同形式的式子算不同解),如没有则输出无解.
算法说明
首先解决图形扑克牌的显示问题.我选择了Qcard.dll.运用其中的DrawCard过程可轻松实现扑克的显示问题,在源程序中会有具体用法.
接下来是24点算法的讨论.首先想到的是用穷举表达式的方法,然后求值.然而,由于括号的存在,使穷举表达式并非易事.实际上,括号的作用仅仅是提高运算的优先级而已,如果我们规定符号的优先级,一样可以达到要求.具体来说,设四张牌为a,b,c,d,运算符为①,②,③,表达式为a ① b ② c ③ .如果强制规定①,②,③的优先顺序,就不必考虑括号问题了.而这3个运算符的运算顺序有3!=6种,分别是:
1.①②③ 2.①③② 3.②①③ 4.②③① 5.③①② 6.③②①
等价的表达式分别是:
1.((a①b②)c③) 2.(a①b)②(c③d) 3.(a①(b②c))③d
4.a①((b②c)③d) 5.(a①b)②(c③d) 6. a①(b②(c③d))
显然,2和5是相同的,因此只考虑5种情况.这样,括号的问题就解决了.
接下来,就是生成a,b,c,d的全排列,注意去掉其中的相同排列.去除的方法很多,比如字典排序等,我用的是另一种方法.
用循环的嵌套生成a,b,c,d的24种全排列,记录在数组中.把每一组数当作一个四位的14进制数,把这24个数全部转化为十进制(如(6529)14=6*143+5*142+2*14+9).这样,如果两个排列完全相同,则得到的十进制数是相等的.这样,通过对这些十进制的比较,就可以比较这些排列的相同情况.一旦遇到相同的排列,就标记上.最后生成一组没有重复的排列.
对这组排列进行以上方法的运算,就可以得到所有的结果了.注意在运算过程中除法的特殊性——除数不能为零.因为可能会用到除法,所以要考虑精度问题,这里通过结果减去24取绝对值与一个接近0的小数比较,如小于它,即可判定结果是24.
附:其他待决的问题:
图形扑克牌的遮挡问题.当窗口中的扑克牌被遮挡后,扑克牌不会重新画上,造成扑克牌遮挡后显示不全问题.应寻找Qcard.dll的有关参数.
形式不同而实质相同的解的问题.有些解虽然形式不同,但其实质是完全相同的.如3*((11+4)-7)和3*(11+(4-7)),实际上只是一种解.去掉这些相同解的问题情况较多,其较为繁琐,有待解决.
多余括号好问题.有些解的括号是多余的,应在输出前去掉.
改进程序的可玩性.增加玩家输入表达式的功能,并判断对错,还可以加上时间限制,使玩家参与到游戏中.
程序框图
VB源程序代码
'需要声明所有用到的变量
Option Explicit
'声明全局变量,数组
Dim cards(1 To 4) As Single, card(1 To 4) As Single
Dim result(1 To 24, 0 To 4) As Integer, final(1 To 24, 1 To 4) As Integer, temp(1 To 24) As Long
Dim nokey As Boolean, total As Integer, n1 As Integer, n2 As Integer, n3 As Integer, n4 As Integer, a As Integer, b As Integer, c As Integer, d As Integer, op1 As Integer, op2 As Integer, op3 As Integer, answer1 As Single, answer2 As Single, answer3 As Single, color As Integer
Dim i As Integer, j As Integer, t As Integer
'声明zero常量,设置0的标准,处理除法的精度问题
Const zero = 0.00001
'初始化QCARD32.DLL
Private Declare Function InitializeDeck Lib "qcard32.dll" (ByVal hwin As Long) As Integer
'DrawCard 子程序,画出扑克牌图样在FORM窗体及 窗体上的图片框
'用法:
'hwnd ---- 需要画图的对象句柄
'nCard --- 扑克牌编号 其编号如下
'1-13 梅花 14-26 方块 27-39 红心 40-52 黑桃 小王-110 大王-111
'x,y 位置
Private Declare Sub DrawCard Lib "qcard32.dll" (ByVal hwnd As Long, ByVal nCard As Integer, ByVal x As Integer, ByVal y As Integer)
'DrawBack 子程序,画出扑克牌的背面图案,共六种 按 1--6 编号
Private Declare Sub DrawBack Lib "qcard32.dll" (ByVal hwnd As Long, ByVal nCard As Long, ByVal x As Long, ByVal y As Long)
'GetCardSuit 函数,求 nCard 的点数 1-13
'Private Declare Function GetCardSuit Lib "qcard32.dll" (ByVal nCard As Long) As Long
'GetCardValue 函数,求 nCard 的花色 0:鬼牌 1:梅花 2:方块 3:红心 4:黑桃
'Private Declare Function GetCardValue Lib "qcard32.dll" (ByVal nCard As Long) As Long
'Form_Load过程,初始化
Private Sub Form_Load()
Randomize Timer
Call InitializeDeck(Me.hwnd)
Command3.Enabled = False
End Sub
'answer函数,返回x与y做operator运算后的值,-100为错误标志
Private Function answer(x As Single, y As Single, operator As Integer) As Single
Select Case operator
Case 1
answer = x + y
Exit Function
Case 2
answer = x - y
Exit Function
Case 3
answer = x * y
Exit Function
Case 4
If y = 0 Then
answer = -100
Exit Function
Else
answer = x / y
Exit Function
End If
End Select
answer = -100
End Function
'operate函数,返回数值op所对应的四则运算符号
Private Function operate(op As Integer) As String
Select Case op
Case 1
operate = "+"
Case 2
operate = "-"
Case 3
operate = "*"
Case 4
operate = "/"
End Select
End Function
'search过程,去掉数组result中相同的元素,存入数组final中
Private Sub search()
For i = 1 To 24
result(i, 0) = 0
temp(i) = result(i, 1) * 14 ^ 3 + result(i, 2) * 14 ^ 2 + result(i, 3) * 14 + result(i, 4)
Next i
For i = 1 To 23
For j = i + 1 To 24
If temp(i) = temp(j) Then result(i, 0) = 1
Next j
Next i
For i = 1 To 24
If result(i, 0) = 1 Then GoTo 1
t = t + 1
For j = 1 To 4
final(t, j) = result(i, j)
Next j
1 Next i
End Sub
'Main过程,用于计算四个数通过不同运算得到24的所有情况,并输出结果
Private Sub Main()
For op1 = 1 To 4
For op2 = 1 To 4
For op3 = 1 To 4
'1·形如( a @ b ) @ c ) @ d 的表达式
answer1 = answer(cards(1), cards(2), op1)
answer2 = answer(answer1, cards(3), op2)
answer3 = answer(answer2, cards(4), op3)
If answer1 -100 And answer2 -100 And answer3 -100 Then
If Abs(answer3 - 24) < zero Then
nokey = False
total = total + 1
Text1.Text = Text1.Text + "((" + Trim$(Str$(cards(1))) + operate(op1) + Trim$(Str$(cards(2))) + ")" + operate(op2) + Trim$(Str$(cards(3))) + ")" + operate(op3) + Trim$(Str$(cards(4))) + " "
'若本行已有三个式子,就换行
If total Mod 3 = 0 Then
Text1.Text = Text1.Text + Chr$(13) + Chr$(10)
End If
End If
End If
'2·形如( a @ b ) @ (c @ d) 的表达式
answer1 = answer(cards(1), cards(2), op1)
answer2 = answer(cards(3), cards(4), op3)
answer3 = answer(answer1, answer2, op2)
If answer1 -100 And answer2 -100 And answer3 -100 Then
If Abs(answer3 - 24) < zero Then
nokey = False
total = total + 1
Text1.Text = Text1.Text + "(" + Trim$(Str$(cards(1))) + operate(op1) + Trim$(Str$(cards(2))) + ")" + operate(op2) + "(" + Trim$(Str$(cards(3))) + operate(op3) + Trim$(Str$(cards(4))) + ")" + " "
'若本行已有三个式子,就换行
If total Mod 3 = 0 Then
Text1.Text = Text1.Text + Chr$(13) + Chr$(10)
End If
End If
End If
'3·形如( a @ ( b @ c ) ) @ d 的表达式
answer1 = answer(cards(2), cards(3), op2)
answer2 = answer(cards(1), answer1, op1)
answer3 = answer(answer2, cards(4), op3)
If answer1 -100 And answer2 -100 And answer3 -100 Then
If Abs(answer3 - 24) < zero Then
nokey = False
total = total + 1
Text1.Text = Text1.Text + "(" + Trim$(Str$(cards(1))) + operate(op1) + "(" + Trim$(Str$(cards(2))) + operate(op2) + Trim$(Str$(cards(3))) + "))" + operate(op3) + Trim$(Str$(cards(4))) + " "
'若本行已有三个式子,就换行
If total Mod 3 = 0 Then
Text1.Text = Text1.Text + Chr$(13) + Chr$(10)
End If
End If
End If
'4·形如 a @ ( ( b @ c ) @ d ) 的表达式
answer1 = answer(cards(2), cards(3), op2)
answer2 = answer(answer1, cards(4), op3)
answer3 = answer(cards(1), answer2, op1)
If answer1 -100 And answer2 -100 And answer3 -100 Then
If Abs(answer3 - 24) < zero Then
nokey = False
total = total + 1
Text1.Text = Text1.Text + Trim$(Str$(cards(1))) + operate(op1) + "((" + Trim$(Str$(cards(2))) + operate(op2) + Trim$(Str$(cards(3))) + ")" + operate(op3) + Trim$(Str$(cards(4))) + ")" + " "
'若本行已有三个式子,就换行
If total Mod 3 = 0 Then
Text1.Text = Text1.Text + Chr$(13) + Chr$(10)
End If
End If
End If
'5·形如 a @ ( b @ ( c @ d ) ) 的表达式
answer1 = answer(cards(3), cards(4), op3)
answer2 = answer(cards(2), answer1, op2)
answer3 = answer(cards(1), answer2, op1)
If answer1 -100 And answer2 -100 And answer3 -100 Then
If Abs(answer3 - 24) < zero Then
nokey = False
total = total + 1
Text1.Text = Text1.Text + Trim$(Str$(cards(1))) + operate(op1) + "(" + Trim$(Str$(cards(2))) + operate(op2) + "(" + Trim$(Str$(cards(3))) + operate(op3) + Trim$(Str$(cards(4))) + "))" + " "
'若本行已有三个式子,就换行
If total Mod 3 = 0 Then
Text1.Text = Text1.Text + Chr$(13) + Chr$(10)
End If
End If
End If
Next op3
Next op2
Next op1
End Sub
'Card1_MouseDown过程,按左键点击纸牌加1,按右键减1
Private Sub Card1_MouseDown(Button As Integer, Shift As Integer, x As Single, y As Single)
Select Case Button
'按左键
Case 1
If card(4) = 13 Then
card(4) = 1
Else
card(4) = card(4) + 1
End If
'按右键
Case 2
If card(4) = 1 Then
card(4) = 13
Else
card(4) = card(4) - 1
End If
End Select
'随机产生变化后的花色
color = Int(Rnd() * 4)
'重画纸牌
Call DrawCard(Me.hwnd, color * 13 + card(4), 10, 10)
End Sub
'Card2_MouseDown过程,按左键点击纸牌加1,按右键减1
Private Sub Card2_MouseDown(Button As Integer, Shift As Integer, x As Single, y As Single)
Select Case Button
'按左键
Case 1
If card(3) = 13 Then
card(3) = 1
Else
card(3) = card(3) + 1
End If
'按右键
Case 2
If card(3) = 1 Then
card(3) = 13
Else
card(3) = card(3) - 1
End If
End Select
'随机产生变化后的花色
color = Int(Rnd() * 4)
'重画纸牌
Call DrawCard(Me.hwnd, color * 13 + card(3), 10 + 85, 10)
End Sub
'Card3_MouseDown过程,按左键点击纸牌加1,按右键减1
Private Sub Card3_MouseDown(Button As Integer, Shift As Integer, x As Single, y As Single)
Select Case Button
'按左键
Case 1
If card(2) = 13 Then
card(2) = 1
Else
card(2) = card(2) + 1
End If
'按右键
Case 2
If card(2) = 1 Then
card(2) = 13
Else
card(2) = card(2) - 1
End If
End Select
'随机产生变化后的花色
color = Int(Rnd() * 4)
'重画纸牌
Call DrawCard(Me.hwnd, color * 13 + card(2), 10 + 2 * 85, 10)
End Sub
'Card4_MouseDown过程,按左键点击纸牌加1,按右键减1
Private Sub Card4_MouseDown(Button As Integer, Shift As Integer, x As Single, y As Single)
Select Case Button
'按左键
Case 1
If card(1) = 13 Then
card(1) = 1
Else
card(1) = card(1) + 1
End If
'按右键
Case 2
If card(1) = 1 Then
card(1) = 13
Else
card(1) = card(1) - 1
End If
End Select
'随机产生变化后的花色
color = Int(Rnd() * 4)
'重画纸牌
Call DrawCard(Me.hwnd, color * 13 + card(1), 10 + 3 * 85, 10)
End Sub
'Command1_Click过程,点击洗牌按钮画出纸牌背面
Private Sub Command1_Click()
'随机产生纸牌背面的样式
color = Int(Rnd() * 6 + 1)
'画出纸牌背面
Call DrawBack(Me.hwnd, color, 10, 10)
Call DrawBack(Me.hwnd, color, 95, 10)
Call DrawBack(Me.hwnd, color, 180, 10)
Call DrawBack(Me.hwnd, color, 265, 10)
'禁用答案按钮
Command3.Enabled = False
End Sub
'Command2_Click过程,点击发牌按钮画出随机产生的纸牌
Private Sub Command2_Click()
'清空答案
Text1.Text = ""
'随机产生的纸牌,并画出
For i = 1 To 4
card(i) = Int(Rnd() * 13 + 1)
color = Int(Rnd() * 4)
Call DrawCard(Me.hwnd, color * 13 + card(i), 10 + (4 - i) * 85, 10)
Next i
'开启答案按钮
Command3.Enabled = True
End Sub
'Command3_Click过程,点击答案按钮计算结果
Private Sub Command3_Click()
'清空解的数量
Label1.Caption = ""
'默认设置为无解
nokey = True
'解的计数器清零
total = 0
'临时变量清零
i = 0
j = 0
t = 0
'产生24种全排列
For n1 = 1 To 4
For n2 = 1 To 4
If n2 = n1 Then GoTo 2
For n3 = 1 To 4
If n3 = n1 Or n3 = n2 Then GoTo 3
n4 = 10 - n1 - n2 - n3
i = i + 1
result(i, 1) = card(n1)
result(i, 2) = card(n2)
result(i, 3) = card(n3)
result(i, 4) = card(n4)
3 Next n3
2 Next n2
Next n1
'调用search过程,去掉重复排列
Call search
'调用Main过程,寻找答案
For i = 1 To t
For j = 1 To 4
cards(j) = final(i, j)
Next j
Call Main
Next i
'输出解的情况
If nokey = False Then Label1.Caption = "共有" + Trim$(Str$(total)) + "组解!" Else Label1.Caption = "无解!"
'禁止答案按钮
Command3.Enabled = False
End Sub
五,程序界面
N
Y
比较temp(i)与temp(j)是否相等
For j=i+1 to 24
For i=1 to 23
初始化变量,数组
For i=1 to 24
初始化重复排列的标志result(1..24,0)=0
排列存入数组result(1..24,1..4)
全排列循环终止
3重循环产生24种全排列
随机产生四张扑克牌
将全排列所对应的十进制数存入temp(1..24)
Next i
设置重复排列的标志result(i,0)=1
计数器t=t+1
Next j
Next i
结束
输出解的个数或无解
结束循环final中的排列
计算形如 a @ ( b @ ( c @ d ) ) 的表达式
如等于24则输出
计算形如 a @ ( ( b @ c ) @ d ) 的表达式
如等于24则输出
计算形如( a @ ( b @ c ) ) @ d的表达式
如等于24则输出
计算形如( a @ b ) @ ( c @ d ) 的表达式
如等于24则输出
计算形如( a @ b ) @ c ) @ d 的表达式
如等于24则输出
结束循环3个运算符
循环3个运算符
循环final中的全部排列
Next i
把result存入final
Y
N
检验result(i,0)标志
是否为1
For i=1 to t