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The 3n + 1 Problem or Collatz Problem

The 3n + 1 Problem, also known as the 3n + 1 Conjecture or Collatz Problem or Collatz Conjecture, is that if any positive integer n is iteratively operated using two simple rules: Addition and Division, then the end-result always leads to the number 1.

The Rules

Rule 1 [Addition]: If the number n is odd, then triple it and add 1.

Rule 2 [Division]: If the number n is even, then divide it by 2.

Steps to Follow

Step 1: Take any positive integer n.

Step 2: Check whether the integer is odd or even.

Step 3: If the integer is odd, then triple it and add 1.

Step 4: If the integer is even, then divide it by 2.

Step 5: Repeat this procedure.

Step 6: The end-result always leads to the number 1.

For Example

Integer n = 5

5 is an odd number, so triple it and add 1. The value of n = 16.

16 is an even number, so divide it by 2. The value of n = 8.

8 is an even number, so divide it by 2. The value of n = 4.

4 is an even number, so divide it by 2. The value of n = 2.

2 is an even number, so divide it by 2. The value of n = 1.

The sequence of the integer 5 is: 5, 16, 8, 4, 2, 1.

As you can see here, the end-result leads to the number 1.

Integer n = 9

9 is an odd number, so triple it and add 1. The value of n = 28.

28 is an even number, so divide it by 2. The value of n = 14.

14 is an even number, so divide it by 2. The value of n = 7.

7 is an odd number, so triple it and add 1. The value of n = 22.

22 is an even number, so divide it by 2. The value of n = 11.

11 is an odd number, so triple it and add 1. The value of n = 34.

34 is an even number, so divide it by 2. The value of n = 17.

17 is an odd number, so triple it and add 1. The value of n = 52.

52 is an even number, so divide it by 2. The value of n = 26.

26 is an even number, so divide it by 2. The value of n = 13.

13 is an odd number, so triple it and add 1. The value of n = 40.

40 is an even number, so divide it by 2. The value of n = 20.

20 is an even number, so divide it by 2. The value of n = 10.

10 is an even number, so divide it by 2. The value of n = 5.

5 is an odd number, so triple it and add 1. The value of n = 16.

16 is an even number, so divide it by 2. The value of n = 8.

8 is an even number, so divide it by 2. The value of n = 4.

4 is an even number, so divide it by 2. The value of n = 2.

2 is an even number, so divide it by 2. The value of n = 1.

The sequence of the integer 9 is: 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

As you can see here, the end-result leads to the number 1.

Hope you have understood this amazing concept of the 3n + 1 Problem or Collatz Problem. You can try yourself using any other positive integer n. In case of any issue, kindly provide your valuable comments.

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