The Collatz Conjecture The 3n 1 Problem

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The Collatz ConjectureThe 3n1 Problem

• Barrett Walls and Iason Rusodimos
• Georgia Perimeter College
• Dunwoody, GA

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The 3n1 Problem

• Pick a positive integer.
• If the number is 1 stop.
• If it is even then divide by 2.
• If it is odd (and gt1) multiply by 3 and add 1.
• Repeat the process.

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• If we start with 6 well get the sequence
• 6,3,10,5,16,8,4,2,1
• If we start with 9 well get
• 9,28,14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2
  ,1
• If we start with 16 well get
• 16,8,4,2,1

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The Collatz Conjecture

• Will you always eventually get to one?
• In 1937 Lothar Collatz proposed the problem and
  conjectured that you will get to one starting
  with any positive integer.
• We still do not know the answer.

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What we do know

• It is true for up some pretty big numbers (Number
  up to 5.764X1018 have been checked with
  computers.)
• This doesnt prove it, though, other conjectures
  have had really large counterexamples (The
  smallest known counterexample of the Mertens
  conjecture is roughly 10(1040))
• There are some interesting patterns that form
  when you look at the sequences.

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• Let counter(n) be the length of the 3n1
  sequence.
• The sequence for 6 is 6,3,10,5,16,8,4,2,1 hence
  counter(6) 9.
• The sequence for 7 is 7, 22, 11, 34, 17, 52,
  26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, hence
  counter(7) 17.
• The function counter(n) is not always increasing,
  counter(8) 4 for example.

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A Graph of counter(n)
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• Let max(n) be the largest number in the 3n1
  sequence.
• The sequence for 6 is 6,3,10,5,16,8,4,2,1 hence
  max(6) 16.
• The sequence for 7 is 7, 22, 11, 34, 17, 52,
  26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, hence
  max(7) 52.
• The function max(n) is not always increasing,
  max(8)8 for example.

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Some graphs of max(n)
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• No sequence for n 1 to n 26 contains a number
  larger than 160.
• The sequence for 27 though gets as high as 9232.
• The sequence for 31 gets to 9232 as well.
• In fact for numbers from n1 to n 100, the only
  times max(n) gt 250 is when it is exactly 9232.

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• For larger ranges even stranger patterns form if
  we look at the maximums

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• Zooming in we see how various patterns of lines
  are formed.

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• Another way to visualize these sequences is to
  arrange the integers on concentric circles.
• The center is one, with powers of two directly
  above it.
• Other numbers are evenly spaced around the circle
  starting with the power of two.

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• To visualize a path we connect the dots.
• For example the sequence for seven is 7, 22,
  11, 34, 17, 52, 26, 13, 40, 20, 10, 5,16, 8, 4,
  2, 1 which gives the picture

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• Here are the graphs for n 25, 26, and 27

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What does this show us?

• Not much about this particular problem, but
  something important about mathematics.
• Problems can look simple but in reality be very
  complex.
• Seemingly random sequences are often full of
  patterns is we look at them in different ways.
• Trying (but not necessarily solving) hard
  problems can lead to interesting results.

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A Few Notes.

• Thank you Gainesville Math Department! (I
  first learned of this problem at the tournament
  here a few years ago.)
• Thanks to Lothar Collatz.
• Thanks to Google and Wikipedia (search for either
  Collatz or 3n1 problem).
• I used Mathematica to create all my pictures, you
  can get the code at
• http//facstaff.gpc.edu/bwalls

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The Bakers Transformation
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• The bakers map is an example of a chaotic map on
  0,1