Mathematical Fun With Happy Numbers

Use Happy Numbers in your math classroom at either elementary or secondary level. A variety of educational uses.

Mathematical Fun With Happy Numbers

(Published in Directory : Mathematics : Fun With Numbers)

When I was in high school, a friend of mine--who happened to also be a foe from a rival math team--introduced me to the concept of a Happy Number.

"A Happy Number," she said, "is a number that, if you square its digits, and add them together, and then take the result and square its digits and add them together, and keep doing that over and over again, you come down to the number 1."

Clear? Well, I didn't get what she was saying until she gave me an example. So, let's try the number 32.

32: 3² + 2² = 9 + 4 = 13
13: 1² + 3² = 1 + 9 = 10
10: 1² + 0² = 1 + 0 = 1

Ah-ha! It comes down to the number 1, so that means 32 is a Happy Number! (It also means that 13 and 10 are Happy Numbers as well, right?)

Well, that raised a question in my mind: How many times do you have to go through that cycle? What if it never comes down to 1? How will you know?

Her answer was simple: If it never comes down to 1, it will come down to 4 instead. (Presumably, that's an UN-happy number!)

So here's an example of an Unhappy Number:

25: 2² + 5² = 4 + 25 = 29
29: 2² + 9² = 4 + 81 = 85
85: 8² + 5² = 64 + 25 = 89
89: 8² + 9² = 64 + 81 = 145
145: 1² + 4² + 5² = 1 + 16 + 25 = 42
42: 4² + 2² = 16 + 4 = 20
20: 2² + 0² = 4 + 0 = 4

She was right! It came down to 4!

Do Happy Numbers have any practical application in the real world? Probably not. But there are some interesting questions and activities that are related to these numbers. Some are for advanced students, some are not. Here are some things you might want to have your students try. The first list is great for students who are struggling with addition facts and multiplication facts. The second list is for more advanced students.

Math Fact Drilling

Is your telephone number a happy number?
Is your street address a happy number?
What about your weight, age, height? Are they happy?
Find all the happy numbers between 1 and 20
When you're going for a ride in the car, check out the license plate numbers. Can you figure out any of them in your head? (This is a challenge, but makes a great travel game.)

Theoretical Problems

Are there an infinite number of happy numbers?
Are there an infinite number of unhappy numbers?
Ellen told me that every number comes down to either one or four--can you prove it? (Hint: I wrote a computer program to help me with this one.)
Can you find any patterns in the sequence of happy umbers?
Does the ratio of happy to unhappy numbers approach a limit?

Finally, I will leave you with a unique number, which is called a Happy Cube. The number is 153. Watch what happens when you do the happy number process, only cubing instead of squaring.

153: 1³ + 5³ + 3³ = 1 + 125 + 27 = 153

As far as I know, 1 and 153 are the only numbers that behave this way--when you sum the cubes of the digits, you get the number you started with. Can you find any others?

About the Author

Name: Douglas Twitchell

Website: http://www.jeorgethedodo.com/dougblog/

Bio: I am the designer, maintainer, and editor of this site (Articles For Educators).

I am a former educator in both elementary and secondary math and science. In addition to this site, I also built and maintain the following educational sites: The Problem Site, Tile Puzzler, and Quote Puzzler.

I'm also a ventriloquist, and you can find more about my puppets here: Jeorge The Dodo

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Selected Member Comments

Below you will find selected member comments about this article. To view all the user comments, please click here: Member Comments Page

Humphrey Nowlin wrote on Feb 25, 2005	quote options
Hi there, I just got done reading your "HAPPY NUMBERS" article, and just wanted to say, you DON'T have all the happy cubes. Of course, zero works, but that's trivial. There are also 370, 371 and 407! Is that all of them?
MartynJohnston wrote on Aug 26, 2005	quote options
There are also several 'Happy (whatever you call something to the power of 4)'. ie. when you sum the fourth power of the digits, you get the number you started with. I reckon that 1 (of course), 1634, 8208, and 9474 are the only ones (I stopped looking beyond 33000).
MartynJohnston wrote on Aug 26, 2005	quote options
I just found a Happy Seventh Power thing. 4,210,818 seems to work. ie. when you sum the seventh power of the digits, you get the number you started with. Are there any others ?

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