The 4-2-1 Loop

This is not a trick. Yet mathematics does provide curiosities that appear to be magic. This is one that has baffled mathematicians for many years and still no one knows why it happens. Try it, you’ll like it!

Pick any whole number and enter it into your calculator.

Ø  If the number is even, divide by 2. If it is odd, multiply by 3 and add 1.

Ø  Repeat the process with the new number over and over. What happens?

Ø  The sequence always ends in the "loop": 4.....2.....1.....4.....2.....1... 

Let’s try it for the arbitrarily selected number 17

17 is odd, so we multiply by 3 and add 1. We get 52. (17×3=51+1=52)

52 is even, so we divide by 2. We get 26. (52 / 2 = 26)

26 is also even, so we divide by 2 again and get 13.

13 is odd, so we multiply by 3 and add 1. We get 40. (13×3=39+1=40)

40 is even, so we divide by 2. We get 20. (40 / 2 = 20)

20 is even, so we divide by 2 and get 10.

10 is also even so we divide by 2 again and get 5.

5 is odd so we multiply by 3 and add 1. We get 16.

16 is even, so we divide by 2 and get 8.

8 is also even so we divide by 2 again and get 4.

4 is even so we divide by 2. We get 2.

2 is even, so we divide by 1 and get 1.

1 is odd, so we multiply by 3 and add 1. We get 4.

4 is even so we divide by 2. We get 2. And so we begin the loop

4.....2.....1.....4.....2.....1...

So, the loop 4..2..1 goes on and on.

It is believed that, no matter which number we begin with (here we started with 17), we will eventually get to 1. This is truly remarkable! Try it for some other numbers to convince yourself that it really does work. Had we started with 12 as our arbitrarily selected number, we would have required 9 steps to reach 1. Starting with 43 will require 29 steps.

Tips: The angel number 421 is the smallest prime formed by the two powers in logical order from right to left.