Happy and Unhappy Numbers Math/Grade 6

Happy and Unhappy Numbers

To determine if a number is happy or not, one must complete a series of steps. Find the sum of the squares of each digit of the original number. Repeat this process with the generated sum. After repeated times of doing this, if the sum is ever 1, then the number is happy. If the sum never is one, then the number is unhappy.

For example, 7 is a happy number. 133 is also a happy number.

72 = 49 12 + 32 + 32 = 19 42 + 92 = 97 12 + 92 = 82 92 + 72 = 130 82 + 22 = 68 12 + 32 + 02 = 10 62 + 82 = 100 12 + 02 = 1 12 + 02 + 02 = 1

1. Is 44 happy or not?

2. Is 35 happy or not?

3. What did you notice about the sums of the squares when working with 35?

PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project Happy and Unhappy Numbers Math/Grade 6

4. Is 73 happy or not? How do you know?

5. The number 23 is a happy number. Based on this, what other numbers do you know would also automatically be happy? How do you know?

6. Sort the following numbers as happy or unhappy: 6, 13, 19, 20, 91, 109, 130, 167

Happy Numbers Unhappy Numbers

7. Did you need to solve for every number in #6? Why or why not?

Happy and Unhappy Numbers - Key

To determine if a number is happy or not, one must complete a series of steps. Find the sum of the squares of each digit of the original number. Repeat this process with the generated sum. After repeated times of doing this, if the sum is ever 1, then the number is happy. If the sum never is one, then the number is unhappy.

For example, 7 is a happy number. 133 is also a happy number.

72 = 49 12 + 32 + 32 = 1 + 9 + 9 = 19 42 + 92 = 16 + 81 = 97 12 + 92 = 1 + 81 = 82 92 + 72 = 81 + 49 = 130 82 + 22 = 64 + 4 = 68 12 + 32 + 02 = 1 + 9 + 0 = 10 62 + 82 = 36 + 64 = 100 12 + 02 = 1 12 + 02 + 02 = 1

1. Is 44 happy or not? Happy

42 + 42 = 32

32 + 22 = 13

12 + 32 = 10

12 + 02 = 1

2. Is 73 happy or not? Unhappy

72 + 32 = 49 + 9 = 58

52 + 82 = 25 + 64 = 89

82 + 92 = 64 + 81 = 145

12 + 42 + 52 = 1 + 16 + 25 = 42

42 + 22 = 16 + 4 = 20

22 + 02 = 4

42 = 16

12 + 62 = 1+ 36 = 37

32 + 72 = 9 + 49 = 58 which is the same sum found on the first line – this will repeat over and over and will never reach a sum of 1.

All numbers will come to a sum of 1 or will enter the infinite loop of 4, 16, 37, 58, 89, 145, 42, 20, 4 at some point.

3. What did you notice about the sums of the squares when working with 73?

See explanation at the bottom of number 2

4. Is 35 happy or not? How do you know?

32 + 52 = 34

32 + 42 = 25

22 + 52 = 29

22 + 92 = 85

82 + 52 = 89

89 is one of the numbers in the infinite loop – therefore, 35 is an unhappy number.

5. The number 23 is a happy number. Based on this, what other numbers do you know would also be happy? How do you know?

All permutations of a happy number are also happy, so 32 is happy. You can also add zeroes to 23 and 32 to make additional happy numbers. For example, 230, 203, 320, 302, 2300, 2003, 2030 etc… are all happy. Student responses may vary.

6. Sort the following numbers as happy or unhappy: 5, 13, 16, 19, 20, 58, 91, 109, 130, 167

Happy Numbers Unhappy Numbers

13, 130, 19, 91, 109, 167 5, 16, 20, 58

7. Did you need to solve for every number in #6? Why or why not?

13 is happy so 130 is also happy

16 and 58 are numbers in the infinite loop of unhappy

Since 91 is happy, then 109 is also happy

PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project