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Mathematical Fun With Happy Numbers

Lesson Plans > Mathematics > Number Theory
 

Mathematical Fun With Happy 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?


Here is an interesting 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

There are other happy cubes;  can you find any?

Here's one last tidbit, found by a reader (Martyn Johnston): 4,210,818 is a happy seventh-power. Why?

47 + 27 + 17 + 07 + 87 + 17 + 87 = 4,210,818

Crazy, huh?

Lesson by Mr. Twitchell

Handouts/Worksheets

Happy Number Worksheet

Determine whether the following are happy numbers. Show your work.

  1. Your telephone number


     
  2. Your street address


     
  3. Your zip code


     
  4. Your school classroom number




     
  5. Your favorite number



     
  6. Your age




     
  7. Find all the happy numbers between 1 and 20.






     

Happy Number Theory

A happy number is a number which, if you add together the squares of its digits, and then do the same process on the result recursively, results in a value of 1. For example, 23 is happy because: 22 + 32 = 13; 12 + 32 = 10; 12 + 02 = 1.

Here are some statements you can attempt to prove or disprove.

  1. Let f(x) be the function by which a number is transformed, summing the squares of its digits. Then there exists a number k for which f(x) < x if x>k.
     
  2. There is an infinite number of happy numbers.

     
  3. There is an infinite number of unhappy numbers (number that never reach 1).

     
  4. All unhappy numbers eventually reach 4 instead of 1.

     
  5. if h(x) = the number of happy numbers less than x, and u(x) = the number of unhappy numbers less than x, then
    h(x)
    u(x)
    approaches a limit as x approaches infinity.

     
  6. There are an infinite number of pairs of consecutive happy numbers (example: 391, 392).

     

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