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Coin Loops

 Published on Sunday, 6th December 2020, 04:00 am; Solved by 237;
Difficulty rating: 30%

Problem 737

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 $\theta_n$, about the line, from the centre of the $(n-1)$th coin then the sum of $\displaystyle\sum_{k=2}^n \theta_k$ 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.