“Keep hitting the square root button of a calculator after entering any positive number, “1” will be the eventual result displayed.”

This is observed by many when playing around with the calculator.

To explain this “phenomena”, we shall show that

There are three cases to consider.

** Case-1 **For since (See “These Are No Jokes”).

* Case-2* For we have (see

*Exercise-1*)

We want to show that

.

To this end, let

It gives

i.e.,

Expanding (see “Double Feature on Christmas Day”), we have

Hence,

Consequently,

From (1-2), we see that

when

i.e.,

Therefore,

* Case-3* For we write

And,

Let

It follows that

Recently, a childhood friend of mine shared with me a discovery:

“Pick any integer greater than 1: If the number is even, divide it by 2; if it’s odd, multiply it by 3 and add 1. Take that new number and repeat the process, again and again. You’ll eventually get 1.”

Fig. 1

Fig. 2

Fig. 3

What my friend came upon is the Collatz conjecture, named after Lothan Collatz, who introduced the idea in 1937. As of today, it is still an unsolved problem: we don’t have a proof showing that the claim is true for *all* numbers, even though it has been verified for every number less than

*Exercise-1 *Show that for This is not a joke.