All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Probability
Never 0 mod 3 (Posted on 2016-03-17) Difficulty: 3 of 5
Let k be a positive integer. Suppose that the integers 1, 2, 3, ...3k, 3k + 1 are written down in random order.

What is the probability that at no time during this process, the sum of the integers that have been written up to that time is divisible by 3?

Source: Putnam competition

No Solution Yet Submitted by Ady TZIDON    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts Values for the first 5 k | Comment 2 of 7 |
The following program counts instances directly by permuting strings of the form "1231231231", with k repetitions of "123" followed by a "1". All permutations are counted.

DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
For k = 1 To 5

s$ = ""
For i = 1 To k
  s = s + "123"
Next
s = s + "1"
h$ = s
tr = 0: hit = 0
Do
   tot = 0
   tr = tr + 1
   good = 1
   For i = 1 To Len(s)
     tot = tot + Val(Mid(s, i, 1))
     If tot Mod 3 = 0 Then good = 0: Exit For
   Next
   If good Then hit = hit + 1
   permute s
   DoEvents
 Loop Until s = h
 Text1.Text = Text1.Text & k & Str(Len(s)) & Str(hit) & "/" & tr & mform(hit / tr, "  0.0000000") & crlf
  
Next k
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

Function mform$(x, t$)
  a$ = Format$(x, t$)
  If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
  mform$ = a$
End Function

For the first five k, given below are k, the length of the string, and the probability in fraction and decimal:

1 4 3/12  0.2500000
2 7 15/210  0.0714286
3 10 84/4200  0.0200000
4 13 495/90090  0.0054945
5 16 3003/2018016  0.0014881


  Posted by Charlie on 2016-03-17 14:39:24
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information