Happy Numbers

Goals

• • Looking for patterns

  • Practice squaring and adding numbers

What To Do

• 1. First explain what a Happy Number is with an example (e.g. 7)

     1. The process to test for a happy number is to square the each digit of a number and then add it together. Repeat the process and if you end up with 1 the number is happy

     2. Start with 7

        1. 7x7 = 49

        2. 4x4 + 9x9 = 16 + 81 = 97

        3. 9x9 + 7x7 = 81 + 49 = 130

        4. 1x1 + 3x3 + 0x0 = 1 + 9 + 0 = 10

        5. 1x1 + 0x0 = 1

        6. so 7 is a happy number

  2. Give students some example to start with. Use numbers that are happy numbers to start with.

  3. Give students a number that is not a happy number, How many time do we have to repeat the process before we know it will not go to 1. Try doing the process 10 times and see what happens.

  4. Part 2 was to work out all the happy number from 1-100. Divide the numbers up into groups of ten and assign it group to one student.

• • 1. The student must then work through each set of ten number to see if any of them are happy numbers.

    2. When this is complete plot the numbers on a 'happy number tree'.

• 5. During the process encourage student to look for patterns in their work.

• • 1. Note from the example above that if 7 is a happy number, then 49, 130 and 10 are also.

    2. Note that if 49 is a happy number then 94 is also a happy number as both sum of squares give the same value

    3. Note that 'unhappy' numbers reduce to a repeating cycle (4->16->37->58->89->145->42->20->4...)

Homework

• • Draw the entire Happy Number Tree for numbers upto 100

  • Draw the "Melancoil" Tree for all the unhappy numbers upto 100

Background Information

• What are Happy Numbers

• The Happy Number Tree and "Melancoil" Tree