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Names in Boxes (Posted on 2017-04-09) Difficulty: 3 of 5
The names of 100 prisoners are placed in 100 wooden boxes, one name to a box, and the boxes are lined up on a table in a room. One by one, the prisoners are led into the room; each may look in at most 50 boxes, but must leave the room exactly as he found it and is permitted no further communication with the others.
The prisoners have a chance to plot their strategy in advance, and they are going to need it, because unless every single prisoner finds his own name all will subsequently be executed.
Find a strategy for them which has probability of success exceeding 30%.

Comment: If each prisoner examines a random set of 50 boxes, their probability of survival is an unenviable 1/2100 ∼ 0.0000000000000000000000000000008. They could do worse—if they all look in the same 50 boxes, their chances drop to zero. 30% seems ridiculously out of reach—but yes, you heard the problem correctly!

No Solution Yet Submitted by Ady TZIDON    
Rating: 3.7500 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(3): Loops | Comment 9 of 10 |
(In reply to re(2): Loops by Jer)

After thinking about it I did see that the box pointed to as well as the original box and the box that points to that (and therefore contains the prisoner's number) all belong to that cycle.

Out of 100,000 trials, the below program found 31,339 successes. This version, because of the line highlighted here:

       If looked(ptr) = 0 Then
         looked(ptr) = 1
       Else
         Exit For
         For i = 1 To 100
           If looked(i) = 0 Then
             ptr = i: Exit For
           End If
         Next
       End If

does not continue beyond a given cycle even if it's shorter than 50.

If that line is commented out, the program restarts outside the cycle already tried. Of course this time part of its 50 has been used up, and it uses only what remains of the 50. The program that always goes the full 50 unless it finds the correct number first finds only 31,317 successes.  It looks like looking beyond the initial cycle has no significant benefit.

DefDbl A-Z
Dim crlf$, box(), looked()


Private Sub Form_Load()
 Form1.Visible = True
 
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 maxtrials = 100000
 Randomize Timer
 For trial = 1 To maxtrials
   ReDim box(100)
   For i = 1 To 100
     found = 0
     Do
       r = Int(Rnd(1) * 100 + 1)
       If box(r) = 0 Then
         box(r) = i
         found = 1
       End If
     Loop Until found
   Next
'   For i = 1 To 100
'     Text1.Text = Text1.Text & Str(box(i))
'   Next
'   Text1.Text = Text1.Text & crlf
   good = 1
   For prisoner = 1 To 100
     DoEvents
     found = 0
     ReDim looked(100)
     ptr = prisoner
     For looking = 1 To 50
       DoEvents
       v = box(ptr)
       If v = prisoner Then
         found = 1: Exit For
       End If
       ptr = v
       If looked(ptr) = 0 Then
         looked(ptr) = 1
       Else
         Exit For
         For i = 1 To 100
           If looked(i) = 0 Then
             ptr = i: Exit For
           End If
         Next
       End If
     Next
     If found = 0 Then
   '    Text1.Text = Text1.Text & "Failed for prisoner " & prisoner & crlf
       Exit For
     End If
   Next
   If found Then
  '   Text1.Text = Text1.Text & "Succeeded" & crlf
     success = success + 1
   End If
 Next trial
 
 Text1.Text = Text1.Text & crlf & success & " done"
  
End Sub


  Posted by Charlie on 2017-04-12 14:41:51
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