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
42 + 92 = 97
92 + 72 = 130
12 + 32 + 02 = 10
12 + 02 = 1
12 + 32 + 32 = 19
12 + 92 = 82
82 + 22 = 68
62 + 82 = 100
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?
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
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
42 + 92 = 16 + 81 = 97
92 + 72 = 81 + 49 = 130
12 + 32 + 02 = 1 + 9 + 0 = 10
12 + 02 = 1
12 + 32 + 32 = 1 + 9 + 9 = 19
12 + 92 = 1 + 81 = 82
82 + 22 = 64 + 4 = 68
62 + 82 = 36 + 64 = 100
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
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 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?
No
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