Happy / Sad Number

A happy number is defined by the following process. 
    1. Starting with any positive integer, 
    2. Replacing the number by the sum of the squares of its digits, 
    3. Repeating the process until the number reach 1. 
    4. If the number reaches 1 at the end of the process, the number is said to be Happy, else, unhappy or sad.

The happy numbers below 500 are:

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496

The happiness of a number is preserved by rearranging the digits, and by inserting or removing any number of zeros anywhere in the number.
The unique combinations of the happy numbers below 1,000 follow (the rest are just rearrangements and/or insertions of zero digits):
1, 7, 13, 19, 23, 28, 44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899

986,543,210 : Greatest happy number with no redundant digits.

1,234,456,789 : Smallest zeroless pandigital happy number.

Golden Ration

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The figure on the right illustrates the geometric relationship. Expressed algebraically:

where the Greek letter phi () represents the golden ratio. Its value is:

.