Leetcode 202. Happy Number

Question:

Write an algorithm to determine if a number is "happy".

A happy number is a number defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers.

Example: 19 is a happy number

12 + 92 = 82 82 + 22 = 68 62 + 82 = 100 12 + 02 + 02 = 1

Explanation:

又是一道数学题。如果就按照题意写算法的话非常麻烦，所以要从数学的角度先简化题目。 根据Top Solution的提示，如果要收敛到1，意味着如果出现了两次同样的值，则表示之后还会出现，会一直循环。如果这个值不是1的话，那这个数就不是happy number。 可是为什么不是happy number的数不可能不重复地进行下去呢？不是happy number的数一定会循环重复么？

我们可以假设，不是happy number的数会无限下去，但是所有的结果只可能在（2，N]中，这里N是一个有限实数。如果无限产生结果，那总有一个时间点会出现重复，也就是loop。

所以一旦出现了loop，我们就停止，去检测结果是不是1。

Code:

class Solution(object):
    def isHappy(self, n):
        """
        :type n: int
        :rtype: bool
        """
        numbers = set()
        while n not in numbers:
            numbers.add(n)
            n = sum(int(i)**2 for i in str(n))
        return n== 1