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Problem 935

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Rolling Square

A square of side length $b<1$ is rolling around the inside of a larger square of side length $1$, always touching the larger square but without sliding.

Initially the two squares share a common corner. At each step, the small square rotates clockwise about a corner that touches the large square, until another of its corners touches the large square. Here is an illustration of the first three steps for $b=\frac{5}{13}$.

For some values of $b$, the small square may return to its initial position after several steps. For example, when $b=\frac{1}{2}$, this happens in $4$ steps; and for $b=\frac{5}{13}$ it happens in $24$ steps.

Let $F(N)$ be the number of different values of $b$ for which the small square first returns to its initial position within at most $N$ steps. For example, $F(6)=4$, with the corresponding $b$ values:
$$\frac{1}{2},2-\sqrt{2},2+\sqrt{2}-\sqrt{2+4\sqrt{2}},8-5\sqrt{2}+4\sqrt{3}-3\sqrt{6},$$
the first three in $4$ steps and the last one in $6$ steps. Note that it does not matter whether the small square returns to its original orientation.
Also $F(100)=805$.

Find $F({10}^8)$.

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翻滚的正方形

一个边长为$b<1$的小正方形在边长为$1$的大正方形内部以始终接触大正方形且不滑动的方式翻滚。

一开始，这两个正方形在一个角上重合。此后每一步，小正方形围绕其接触大正方形的一个角顺时针旋转，直到它的另一个角接触到大正方形。如下图所示是$b=\frac{5}{13}$时前三步的示意图。

对于$b$的某些取值，小正方形可能在若干步后回到其初始位置。例如，当$b=\frac{1}{2}$时，需要$4$步；而对于$b=\frac{5}{13}$，则需要$24$步。

令$F(N)$为使小正方形在最多$N$步内首次回到其初始位置的$b$的不同取值的数目。例如，$F(6)=4$，对应的 $b$ 值为：
$$\frac{1}{2},2-\sqrt{2},2+\sqrt{2}-\sqrt{2+4\sqrt{2}},8-5\sqrt{2}+4\sqrt{3}-3\sqrt{6},$$
其中，前三个需要$4$步，最后一个需要$6$步。注意这里并不要求小正方形回到其初始方向。
此外，$F(100)=805$。

求$F({10}^8)$。

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