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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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re(3): computer discovery | Comment 5 of 9 |
(In reply to re(2): computer discovery by Charlie)

Coding for a strategy where you can choose your second card knowing the result of the first within that round:

2  8/3 = 2.66666666666667
3  65/15 = 4.33333333333333
4  622/105 = 5.92380952380952
5  7137/945 = 7.55238095238095
6  95244/10395 = 9.16248196248196
7  1456653/135135 = 10.779242979243
8  25120890/2027025 = 12.3929847929848
9  482692725/34459425 = 14.0075675958029

Comparing with previously mentioned (exceptional) strategy for n=2:

1/3 probability of immediate match and total number of rounds is 2.

If no match on first round (probability 2/3), first card on second round will definitely match one of the first two cards, so there are a total of 3 rounds.

Average is 2.6666...



    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
           
           'new code for non-simultaneous flips
           If vp = 0 Then  ' first choice is the already known value
             removals = removals + 1
             Mid(kn$, psn(v, 1), 1) = Mid(bd$, psn(v, 1), 1)
             aknown = aknown - 1
             chos=chos+1  '  making second choice now
           End If
           
           
         End If
         chos = chos + 1
         If chos = 2 Then Exit For
       End If
      Next
    End If


  Posted by Charlie on 2017-06-01 10:32:45
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