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.
very Nice !!
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