Problem 5

Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to this algorithm: Larry makes a 90° left turn after every

\ell

kilometer driving from the start; Rob makes a 90° right turn after every

r

kilometer driving from the start, where

\ell

and

r

are relatively prime positive integers. If both turns occur simultaneously, the car keeps going without changing direction. Assume the ground is flat and the car can move in any direction. The car starts from Argovia facing towards Zillis. For which pairs

(\ell,r)

is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?

Step 2 of 6: Exclude unequal residues

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\ell-r\equiv2,1,3\pmod4\Longrightarrow\text{bounded path}

Detailed analysis

If $\ell-r\equiv2\pmod4$ , the direction reverses after one section and the second section returns to the starting point. If $\ell-r\equiv1$ or $3\pmod4$ , the direction cycles through four orientations and the car returns to the starting point after four sections. In all these cases the path stays within a bounded distance, so it cannot reach destinations arbitrarily far away.