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Problem 92

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Square Digit Chains

A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.

For example,

$$\begin{array}{rl}& 44\to 32\to 13\to 10\to \mathbf{\text{1}}\to \mathbf{\text{1}}\\ & 85\to \mathbf{\text{89}}\to 145\to 42\to 20\to 4\to 16\to 37\to 58\to \mathbf{\text{89}}\end{array}$$

Therefore any chain that arrives at $1$ or $89$ will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at $1$ or $89$.

How many starting numbers below ten million will arrive at $89$?

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平方数链

从任意一个数开始，不断取其各位数字的平方和，直到出现重复，就得到了一条数链。

例如：

$$\begin{array}{rl}& 44\to 32\to 13\to 10\to \mathbf{\text{1}}\to \mathbf{\text{1}}\\ & 85\to \mathbf{\text{89}}\to 145\to 42\to 20\to 4\to 16\to 37\to 58\to \mathbf{\text{89}}\end{array}$$

如上所示，只要数链中出现$1$或$89$，之后就会进入循环。最令人惊奇的是，从任意一个数开始，最终都必定会到达$1$或$89$。

从任意小于一千万的数开始，有多少个最终会到达$89$？

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