A robot moves in a series of one-fifth circular arcs ($72^\circ$), with a free choice of a clockwise or an anticlockwise arc for each step, but no turning on the spot.
One of $70932$ possible closed paths of $25$ arcs starting northward is
Given that the robot starts facing North, how many journeys of $70$ arcs in length can it take that return it, after the final arc, to its starting position?
(Any arc may be traversed multiple times.)
The robot is moving along the complex plane with the steps being represented by powers of a fifth root of unity. We can calculate the $k$th possible past trip $W_k$ simply by taking the $k$th power of $(1 + w + w^2 + w^3 + w^4)$, where $w = cis(72^{\circ})$. The coefficient of $w^{0}$ (which corresponds to paths finishing to the North) in $W_{70}$ gives the solution.
Using a symbolic computation engine, such as Mathematica or Wolfram Alpha, we get that the solution is `890976376292221107991974`.
So, the robot can take `890976376292221107991974` journeys of $70$ arcs in length that return it, after the final arc, to its starting position.
More Answers:
Dice GameConcealed Square
Integer Partition Equations
Error 403 The request cannot be completed because you have exceeded your quota. : quotaExceeded