Problem 737

Coin Loops

A game is played with many identical, round coins on a flat table.

Consider a line perpendicular to the table.
The first coin is placed on the table touching the line.
Then, one by one, the coins are placed horizontally on top of the previous coin and touching the line.
The complete stack of coins must be balanced after every placement.

The diagram below [not to scale] shows a possible placement of $8$ coins where point $P$ represents the line.

It is found that a minimum of $31$ coins are needed to form a coin loop around the line, i.e. if in the projection of the coins on the table the centre of the $n$ th coin is rotated $θ_{n}$ , about the line, from the centre of the $(n - 1)$ th coin then the sum of $\sum_{k = 2}^{n} θ_{k}$ is first larger than $360^{\circ}$ when $n = 31$ . In general, to loop $k$ times, $n$ is the smallest number for which the sum is greater than $360^{\circ} k$ .

Also, $154$ coins are needed to loop two times around the line, and $6947$ coins to loop ten times.

Calculate the number of coins needed to loop $2020$ times around the line.

硬币环绕

如下游戏需要一张平整的桌子和许多相同大小的圆形硬币。

想象在桌面上有一条竖直的线。
紧贴竖直线，在桌面上平放第一枚硬币。
接着，不断地在上一枚硬币上平放一枚硬币，新的硬币也必须紧贴竖直线。
在每次放上新的硬币时，堆叠的硬币必须始终保持平衡。

如下图展示了摆放 $8$ 枚硬币的俯视示意图（并未精确绘制，仅供参考），其中点 $P$ 表示竖直线。

已知，为了沿着竖直线让硬币环绕一整圈，至少需要 $31$ 枚硬币，也就是说，在俯视图中，记第 $n$ 枚硬币的圆心相对第 $n - 1$ 枚硬币绕着竖直线转过的角度为 $θ_{n}$ ，则当 $n = 31$ 时， $\sum_{k = 2}^{n} θ_{k}$ 第一次大于 $360^{\circ}$ 。一般地，说需要 $n$ 枚硬币以环绕竖直线 $k$ 圈，表示 $n$ 是使上述求和第一次大于 $360^{\circ} k$ 的整数。

又已知，环绕竖直线两圈需要 $154$ 枚硬币，环绕竖直线十圈需要 $6947$ 枚硬币。

求环绕竖直线 $2020$ 圈需要的硬币数目。