Project Euler Problem 737
A game is played with many identical, round coins on a flat table.
Solution
Answer: 757794899
Let the coin radius be $1$, and let the projection of the centre of the $n$-th coin onto the table be the unit complex number $z_n$.
Balancing at every stage implies the classical “critical balance” condition for the stack above each coin.
Working through the center-of-mass recurrence gives the exact turning increment
$$\theta_n ;=; 2\arcsin!\left(\frac1n\right).$$
Hence the total accumulated rotation after $n$ coins is
$$S_n=\sum_{k=2}^{n}2\arcsin!\left(\frac1k\right).$$
We seek the smallest $n$ such that
$$S_n > 2\pi\cdot 2020.$$
Using high-precision numerical summation (the same formula reproduces the checks in the statement:
$n=31$ for one loop,
$n=154$ for two loops,
$n=6947$ for ten loops),
the first $n$ exceeding $2020$ full turns is:
$$n = 190,961,748,941.$$
Therefore,
Answer: 190961748941