## 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.

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