Happy Numbers

Date: 01/05/2006 at 18:05:04
From: margie
Subject: What is a "happy" number?

What is a "happy" number?  What are all the two-digit "happy" numbers
between one and one hundred that are multiples of four?  I have never
heard the term.

Date: 01/06/2006 at 00:55:39
From: Doctor Minter
Subject: Re: What is a 

Hi Margie!

A "happy number" is such that when you square each of its digits, add 
those squares, then repeat the process, eventually you will get an 
answer of 1.

An example of a happy number is 7.  Let's go through the process to 
see why 7 is considered "happy."

   7^2 = 49

   4^2 + 9^2 = 97

   9^2 + 7^2 = 130

   1^2 + 3^2 + 0^2 = 10

   1^2 + 0^2 = 1

The fact that we ended up at 1 means that 7 is a happy number.  Each 
number that we obtained along the way (49,97,130,10) are also happy 
numbers because we could have started the steps at any one of those 
numbers and gotten the same end result, as well as the same 
intermediate steps.  

The opposite is also true; that is, applying these steps to any 
number that is "unhappy" will only yield other unhappy numbers.

The way to tell if a number is unhappy is when you keep applying the 
steps and find a repeated number.  Once you find a repeated number, 
the process will repeat forever.

Why these numbers are called "happy numbers" is beyond me.  Maybe 
they ran out of terms, and someone was in a good mood when they 
thought this up.  Not that I have any idea who "they" are.  I 
digress...

The commutative property of addition implies that any combination of 
digits of a (un)happy number will produce another (un)happy number.  
Therefore, since the above example showed that 49 and 97 are happy 
numbers, reversing their digits will show that 94 and 79 are happy as 
well.  Also, since 64 is an unhappy number (try it!), 46 is unhappy 
as well.

To find all of the happy numbers less than 100 that are multiples of 
4, I advise first writing all multiples of 4 that are less than 100, 
and then evaluate and imply happiness or unhappiness of the others by 
the properties that I have mentioned.

I mentioned that 64 is unhappy, so 8 is also, since applying one step 
to the number 8 will give a result of 64.

I hope that this is enough to get you started on the specific 
problem.  Please feel free to write again if you need further 
assistance or if you have any other questions.  Thanks for using Dr. 
Math!

- Doctor Minter, The Math Forum
  http://mathforum.org/dr.math/