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Tracing a Collatz sequence

mark

(10 points)

The Collatz conjecture is one of the most famous unsolved problems in Mathematics. It’s quite easy to state and understand but it has eluded proof for 85 years. Here’s the conjecture:

Given a positive integer a_0, define a sequence (a_i)_{i=0}^{\infty} recursively as follows:
First, a_0 is already chosen. Then, for i>0, let

a_i = \begin{cases} 3 a_{i-1} + 1 & \text{if }a_{i-1} \text{ is odd} \\ a_{i-1}/2 & \text{if }a_{i-1} \text{ is even}. \end{cases}

The conjecture asks if this sequence always lands on the number 1.

Here’s an illustration of the orbit of 3, for example:

3 3 10 10 3->10 5 5 16 16 5->16 10->5 8 8 16->8 4 4 8->4 2 2 4->2 1 1 2->1 1->4

You should be able to check these values easily enough.

Your problem:

Use the table below to find an a_0 to generate the resulting Collatz sequence. Then respond to this question with

  1. The resulting sequence and
  2. The number of multiplications or divisions required to generate the sequence - until it reaches 1.
  3. The number of additions required to generate the sequence.
  4. A brief explanation of how you got your results.
name a_0
Amanda 5
Anna 6
Benjamin 10
Christian 12
Elida 13
Jared 20
Kasey 21
Matthew 24
Nicholas 26
Patt 32
Patt

Starting number: 32
Sequence: 32\to16\to8\to4\to2\to1
Operations: 5 divisions, 0 multiplications, and 0 additions
Explanations: I realized that 32 = 2^5 so I would have 5 division steps to get to 1.

athach

Starting #: 5

5 \stackrel{3(5)+1}{\longrightarrow} 16 \stackrel{16/2}{\longrightarrow} 8\stackrel{8/2}{\longrightarrow} 4 \stackrel{4/2}{\longrightarrow} 2 \stackrel{2/2}{\longrightarrow} 1
  1. Sequence: 5 , 16, 8, 4, 2, 1
  2. 4 divisions, 1 multiplications
  3. 1 additions
  4. I started out with an odd number, 5, and went from there. Since it was odd I followed the operation stated, which was 3(n)+1, and for the last few numbers, it all turned out to be even so I followed the even operation, (n)/2.
ksimmon1

Starting number: 21

1:

21 \stackrel{3(21) + 1}{\longrightarrow} 64 \stackrel{64/2}{\longrightarrow} 32 \stackrel{32/2}{\longrightarrow} 16 \stackrel{16/2}{\longrightarrow} 8 \stackrel{8/2}{\longrightarrow} 4 \stackrel{4/2}{\longrightarrow} 2 \stackrel{2/2}{\longrightarrow} 1

2: Multiplication: 1 Division: 6

3: Addition: 1

4: I started with an odd number 21, so I followed the operation 3n+1 where the result was 64, which then I followed the operation n/2. Every number after using that operation was also divisible by 2 so it continued to result in even numbers until I reached 1.

athach1
  • 1.) Starting Number: 6
  • 2.) Multiplications: 2. Divisions: 6.
  • 3.) Addition: 2.
  • 4.) I utilized the conjecture of 3n +1 if n was odd and 1/2n if n was even, up until the remaining number which was one.
6 \stackrel{6/3}{\longrightarrow} 3 \stackrel{3\times3+1}{\longrightarrow} 10 \stackrel{10/2}{\longrightarrow} 5 \stackrel{3\times5+1}{\longrightarrow} 16 \stackrel{16/2}{\longrightarrow} 8 \stackrel{8/2}{\longrightarrow} 4 \stackrel{4/2}{\longrightarrow} 2 \stackrel{2/1}{\longrightarrow} 1.
nhowe

Starting Number: 26
1.)

26 \stackrel{26/2}{\longrightarrow} 13 \stackrel{13\times3+1}{\longrightarrow} 40 \stackrel{40/2}{\longrightarrow} 20 \stackrel{20/2}{\longrightarrow} 10 \stackrel{10/2}{\longrightarrow} 5 \stackrel{5\times3+1}{\longrightarrow} 16 \stackrel{16/2}{\longrightarrow} 8 \stackrel{8/2}{\longrightarrow} 4 \stackrel{4/2}{\longrightarrow} 2 \stackrel{2/2}{\longrightarrow} 1

2.) Multiplications: 2 Divisions: 8
3.) Additions: 2
4.) I utilized the conjecture, dividing by 2 whenever the number was even, and multiplying (3x+1) when the number was odd.

Eli

Starting Number: 13
Sequence: 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
Operations: 7 divisions, 2 multiplications, 2 additions
Explanation: I started with 13, which is odd, so I completed the formula 3x+1, this brought me to 40. Through 2 divisions, I arrived at 10, which I recognized as a part of the branch shown in the example. I checked following the 3x+1 if odd x/2 if even and got to 1.

Jared

Starting Number 20
Sequence: 20–10–5–16–8–4–2–1
Operations: 5 divisions, 1 multiplications
Additions: 1
4. Starting with an even number and using the (x)/2 operation given, I divided by 2 down to 5 , then used the 3(x) +1 operation for 5 to get to 16 and divided by 2 down to 1.

csluder2

Starting number : 12
1.Sequence: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1
2. Multiplications: 2, Divisions: 7
3. Additions: 2
4. I started with an even number, which was 12, and worked my way down. used the x/2 operation to get it to 6, then again to 3. Then I used the 3(x)+1 operation for my first multiplication and addition to get 10. Then used the even operation (x/2) again to get 5, then the odd (3(x)+1) to get 16, then the even for the rest of the problem from 16 down to 1.

mark