A Happy Number is any whole number that reduces to the number one after repeatedly applying a simple formula.

Take any digit, and then square it. If the result contains multiple digits, repeat for each digit and add both results together. Continue this process.

If you end up with the number 1, then the original number is a happy number!

For example, let’s take the number 7.

(i) 7 x 7 = 49

(ii) 49 = 4 x 4 + 9 x 9 = 16 + 81 = 97

(iii) 97 = 9 x 9 + 7 x 7 = 81 + 49 = 130

(iv) 130 = 1 x 1 + 3 x 3 + 0 x 0 = 1 + 9 + 0 = 10

(v) 10 = 1 x 1 + 0 x 0 = 1 + 0 = 1

Note that the following numbers are also happy:

49 & 94: 97 & 79: 130 & 13 & 31: 10 & 1

Why?

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Solution

The reversal of the digits of a happy number is also a happy number.

Consider the number 18.

(i) 1 x 1 + 8 x 8 = 1 + 64 = 65

(ii) 65 = 6 x 6 + 5 x 5 = 36 + 25 = 61

(iii) 61 = 6 x 6 + 1 x 1 = 36 + 1 = 37

(iv) 37 = 3 x 3 + 7 x 7 = 9 + 49 = 58

(v) 58 = 5 x 5 + 8 x 8 = 25 + 64 = 89

(vi) 89 = 8 x 8 + 9 x 9 = 64 + 81 = 145

(vii) 145 = 1 x 1 + 4 x 4 + 5 x 5 = 42

(viii) 42 = 4 x 4 + 2 x 2 = 16 + 4 = 20

(ix) 20 = 2 x 2 + 0 x 0 = 4

(x) 4 = 4 x 4 = 16

Note that 18 is not a happy number. It is a sad number…

Why?

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Solution

Reversing the digits of the number 16 produces 61. The number 61 has already appeared earlier after applying the formula. A continuous loop would then occur.

Write down all of the other two digit numbers, which are sad, based on the above calculations.

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Solution

81, 56 & 65, 16 & 61, 37 & 73, 58 & 85, 89 & 98, 24 & 42

List all happy numbers in the first 100 natural numbers.

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Solution

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100

Here are a few observations that we can make about this list.

  1. There are 20 happy numbers in the first 100 natural numbers, or one fifth.
  2. Of these, seven are prime numbers: 7, 13, 19, 23, 31, 79 & 97
  3. There is only one pair of consecutive happy numbers: 31 & 32.
  4. There are three perfect square happy numbers: 1, 49 & 100.