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Memory Match Game (Posted on 2017-05-31) Difficulty: 3 of 5
10 cards with identical backs are face down on a table. Each card face matches exactly one of the other card faces. The cards are in a random sequence. A turn consists of choosing 2 cards, simultaneously reversing them so that they are face up, discarding them if they match, and turning them face down if they do not match. The game ends when all cards are discarded.

a) If you have perfect memory, and an efficient strategy, then what is the expected number of turns in the 10 card game?

b) What is the expected number of turns if instead there are 2n cards in the starting tableaux, with each card matching exactly one other?

See The Solution Submitted by Steve Herman    
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Solution computer discovery | Comment 1 of 9
The following is based on the strategy of going for previously unexposed cards unless two previously exposed cards match, in which case, choose them.  The order in which the cards are sorted determines the sequence they will be selected as they are selected left to right following the rules of choice.

The ten card game: n = 5 in the below table, so the expected value is 9 + 4/9.

For various n:

n   cards   expected turns
2     4    20/6 = 3.33333333333333
3     6    486/90 = 5.4
4     8    18720/2520 = 7.42857142857143
5    10    1071000/113400 = 9.44444444444444
6    12    85730400/7484400 = 11.4545454545455

The pattern seems to be the expected turns = 
  (2*n-1) + (n-1)/(2*n-1)
  
The case for n=7 was done with a variation of the program, that used only combinations starting with 1, as the identity of the individual types does not matter:

7  1309770000/97297200 = 13.4615384615385

The expected number of rounds is indeed 13 + 6/13.

A third variation of the program, using only combinations where the first item is 1, and each successive newly found item is the next numerically, such as 1213324554, speeding things considerably, finds

8  31351320/2027025 = 15.4666666666667
9  602026425/34459425 = 17.4705882352941


The table is from:

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For n = 2 To 6: ntimes2 = 2 * n
 
 totchoices = 0
 DoEvents
 Tr = 0
bd0$ = "112233445566"
bd0$ = Left(bd0, ntimes2)
h$ = bd0$
trials = 0
Do
 If bd0$ = "1122334545" Then
    xx = xx
 End If
  bd$ = bd0$
  kn$ = String$(ntimes2, " ")
  ReDim psn(n, 2)
  mvs = 0
  removals = 0
  aknown = 0
    trials = trials + 1
    choices = 0
  Do
    DoEvents
    p$ = l$       ' just for debugging permutations
    l$ = Left$(bd$, 9)  'just for debugging permutations
    choices = choices + 1
    'If l$ <> p$ Then Text1.Text = Text1.Text & bd$ & crlf
    If aknown Then
      For i = 1 To n
        If psn(i, 2) > 0 And psn(i, 0) = 0 Then
          p1 = psn(i, 1)
          p2 = psn(i, 2)
          removals = removals + 1
          psn(i, 0) = 1
          aknown = aknown - 1
          Exit For
        End If
      Next
    Else
      chos = 0
      vp = 0
      v = 0
      For i = 1 To ntimes2
       If Mid$(kn$, i, 1) = " " Then
         Mid$(kn$, i, 1) = Mid$(bd$, i, 1)
         vp = v
         v = Val(Mid$(kn$, i, 1))
         If vp = v Then
           removals = removals + 1
           psn(v, 0) = 1
           aknown = aknown - 1  'subtracting before adding this time
         End If
         If psn(v, 1) = 0 Then
           psn(v, 1) = i
         Else
           psn(v, 2) = i
           aknown = aknown + 1
         End If
         chos = chos + 1
         If chos = 2 Then Exit For
       End If
      Next
    End If
  Loop Until removals = n
  totchoices = totchoices + choices
  permute bd0$
Loop Until bd0$ = h$

Text1.Text = Text1.Text & n & "  " & totchoices & "/" & trials & " = " & totchoices / trials & crlf

Next n
End Sub




Latest, fastest, version (labels are such that first appearances are in numeric order) :

DefDbl A-Z
Dim crlf$, choices, trials, bd0$, n, totchoices, numOf()


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 

For n = 3 To 9
  totchoices = 0
  DoEvents
  trials = 0: totchoices = 0
  ReDim numOf(n)
  numOf(1) = 1
  bd0$ = "1"
  addOn
  Text1.Text = Text1.Text & n & "  " & totchoices & "/" & trials & " = " & totchoices / trials & crlf

Next n


End Sub

Sub addOn()
 If Len(bd0$) = 2 * n Then
  findchoices bd0$, n
  totchoices = totchoices + choices
 Else
  For i = 1 To n
    If numOf(i) < 2 Then
     If numOf(i) > 0 Or didzero = 0 Then
      If numOf(i) = 0 Then didzero = i
      bd0$ = bd0$ + LTrim(Str(i))
      numOf(i) = numOf(i) + 1
        
      addOn
        
      numOf(i) = numOf(i) - 1
      bd0$ = Left(bd0$, Len(bd0$) - 1)
     End If
    End If
  Next
 End If
End Sub


Sub findchoices(bd$, n)
  ntimes2 = 2 * n
  kn$ = String$(ntimes2, " ")
  ReDim psn(n, 2)
  mvs = 0
  removals = 0
  aknown = 0
    trials = trials + 1
    choices = 0
  Do
    DoEvents
    p$ = l$       ' just for debugging permutations
    l$ = Left$(bd$, 9)  'just for debugging permutations
    choices = choices + 1
    'If l$ <> p$ Then Text1.Text = Text1.Text & bd$ & crlf
    If aknown Then
      For i = 1 To n
        If psn(i, 2) > 0 And psn(i, 0) = 0 Then
          p1 = psn(i, 1)
          p2 = psn(i, 2)
          removals = removals + 1
          psn(i, 0) = 1
          aknown = aknown - 1
          Exit For
        End If
      Next
    Else
      chos = 0
      vp = 0
      v = 0
      For i = 1 To ntimes2
       If Mid$(kn$, i, 1) = " " Then
         Mid$(kn$, i, 1) = Mid$(bd$, i, 1)
         vp = v
         v = Val(Mid$(kn$, i, 1))
         If vp = v Then
           removals = removals + 1
           psn(v, 0) = 1
           aknown = aknown - 1  'subtracting before adding this time
         End If
         If psn(v, 1) = 0 Then
           psn(v, 1) = i
         Else
           psn(v, 2) = i
           aknown = aknown + 1
         End If
         chos = chos + 1
         If chos = 2 Then Exit For
       End If
      Next
    End If
  Loop Until removals = n

End Sub

  Posted by Charlie on 2017-05-31 16:20:20
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