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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,

$$
\begin{aligned}
& 44 \rightarrow 32 \rightarrow 13 \rightarrow 10 \rightarrow \textbf{1} \rightarrow \textbf{1}\\
& 85 \rightarrow \textbf{89} \rightarrow 145 \rightarrow 42 \rightarrow 20 \rightarrow 4 \rightarrow 16 \rightarrow 37 \rightarrow 58 \rightarrow \textbf{89}
\end{aligned}
$$

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：

$$
\begin{aligned}
& 44 \rightarrow 32 \rightarrow 13 \rightarrow 10 \rightarrow \textbf{1} \rightarrow \textbf{1}\\
& 85 \rightarrow \textbf{89} \rightarrow 145 \rightarrow 42 \rightarrow 20 \rightarrow 4 \rightarrow 16 \rightarrow 37 \rightarrow 58 \rightarrow \textbf{89}
\end{aligned}
$$

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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