Programming can do all but

A number is said to be happy if the sum of the squares of the digits of the number is 1.

For example: $28=2^{2}+8^{2}=68=6^{2}+8^{2}=100=1^{2}+0^{2}+0^{2}=1.$ We can square the resultant number as many times until it becomes 1 or less than 10.

Hence 28 is a happy number.

How many happy numbers are there under 100(including 100)?

Note:Can you provide a mathematical solution for this?