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The sum of the squares of the digits of 13 is 12 + 32 = 10. The sum of the squares of the digits of this result is 12 + 02 = 1. Because this process results in a 1, this number is a happy number.
It is somewhat complicated; so I want to point out to the Wikipedia article that explains it: http://en.wikipedia.org/wiki/Happy_numberThe origin of the name "happy" is not explained there, it seems;...