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Here is a general solution. Yeah, there are more than two.

Each city can be said oriented clockwise or counter-clockwise in the sense of which way the blue road goes.

The simple rule is that the cities must be oriented symmetrically relative to Rome. If the city n steps clockwise of Rome is oriented clockwise, then the city n steps counter-clockwise from Rome must be oriented counter-clockwise. And vice-versa.
The path to write down is just whatever it takes to go from the city just left or right of Rome, around the circle, back to Rome.

This give 64 solutions with a path of length 10. You can choose the correlated orientation of the 5 pairs of cities and the orientation of Rome.

It works because ...

Imagine two police cars starting from Rome's two neighbor cities, going around the circle and back to Rome, one clockwise, the other counter-clockwise. They follow the prescribed path.

A car C starting from any other cities at the same time would move randomly at first, following a path for cities it wasn't meant for.
But sooner or later C will meet one of the police cars. And when that happens it will be in the right city at the right time for the path, which means it will then follow the path with the police car down to to Rome.

Depending on the parity of the starting position, one of the police cars will catch the car. If the car tries to escape it and pass Rome before the police car reaches it, it won't help because it will then run into the other police car arriving in the other direction. So, whatever the car does, it will bump into a police car and be directed to Rome.

How general is it?

These are indeed the only solutions for a minimal path length.

It has been shown that the minimum path length is 10. It means a car starting left or right of Rome must go straight around the circle, without changing direction. Since for both a clockwise and a counter-clockwise round trip the instructions are the same, the orientation of the cities must be symmetrical about Rome. And the round trip determines the driving instructions uniquely.
So this condition of symmetry is necessary and we have seen it is also sufficient.

Here is a general solution. Yeah, there are more than two.

Each city can be said oriented clockwise or counter-clockwise in the sense of which way the blue road goes.

The simple rule is that the cities must be oriented symmetrically relative to Rome. If the city n steps clockwise of Rome is oriented clockwise, then the city n steps counter-clockwise from Rome must be oriented counter-clockwise. And vice-versa.
The path to write down is just whatever it takes to go from the city just left or right of Rome, around the circle, back to Rome.

This give 64 solutions with a path of length 10. You can choose the correlated orientation of the 5 pairs of cities and the orientation of Rome.

It works because ...

Imagine two police cars starting from Rome's two neighbor cities, going around the circle and back to Rome, one clockwise, the other counter-clockwise. They follow the prescribed path.

A car C starting from any other cities at the same time would move randomly at first, following a path for cities it wasn't meant for.
But sooner or later C will meet one of the police cars. And when that happens it will be in the right city at the right time for the path, which means it will then follow the path with the police car down to to Rome.

Depending on the parity of the starting position, one of the police cars will catch the car. If the car tries to escape it and pass Rome before the police car reaches it, it won't help because it will then run into the other police car arriving in the other direction. So, whatever the car does, it will bump into a police car and be directed to Rome.

Here is a general solution. Yeah, there are more than two.

Each city can be said oriented clockwise or counter-clockwise in the sense of which way the blue road goes.

The simple rule is that the cities must be oriented symmetrically relative to Rome. If the city n steps clockwise of Rome is oriented clockwise, then the city n steps counter-clockwise from Rome must be oriented counter-clockwise. And vice-versa.
The path to write down is just whatever it takes to go from the city just left or right of Rome, around the circle, back to Rome.

This give 64 solutions with a path of length 10. You can choose the correlated orientation of the 5 pairs of cities and the orientation of Rome.

It works because ...

Imagine two police cars starting from Rome's two neighbor cities, going around the circle and back to Rome, one clockwise, the other counter-clockwise. They follow the prescribed path.

A car C starting from any other cities at the same time would move randomly at first, following a path for cities it wasn't meant for.
But sooner or later C will meet one of the police cars. And when that happens it will be in the right city at the right time for the path, which means it will then follow the path with the police car down to to Rome.

Depending on the parity of the starting position, one of the police cars will catch the car. If the car tries to escape it and pass Rome before the police car reaches it, it won't help because it will then run into the other police car arriving in the other direction. So, whatever the car does, it will bump into a police car and be directed to Rome.

How general is it?

These are indeed the only solutions for a minimal path length.

It has been shown that the minimum path length is 10. It means a car starting left or right of Rome must go straight around the circle, without changing direction. Since for both a clockwise and a counter-clockwise round trip the instructions are the same, the orientation of the cities must be symmetrical about Rome. And the round trip determines the driving instructions uniquely.
So this condition of symmetry is necessary and we have seen it is also sufficient.

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answered yesterday

Florian F

37.3k
4
90
173

Here is a general solution. Yeah, there are more than two.

Each city can be said oriented clockwise or counter-clockwise in the sense of which way the blue road goes.

The simple rule is that the cities must be oriented symmetrically relative to Rome. If the city n steps clockwise of Rome is oriented clockwise, then the city n steps counter-clockwise from Rome must be oriented counter-clockwise. And vice-versa.
The path to write down is just whatever it takes to go from the city just left or right of Rome, around the circle, back to Rome.

This give 64 solutions with a path of length 10. You can choose the correlated orientation of the 5 pairs of cities and the orientation of Rome.

It works because ...

Imagine two police cars starting from Rome's two neighbor cities, going around the circle and back to Rome, one clockwise, the other counter-clockwise. They follow the prescribed path.

A car C starting from any other cities at the same time would move randomly at first, following a path for cities it wasn't meant for.
But sooner or later C will meet one of the police cars. And when that happens it will be in the right city at the right time for the path, which means it will then follow the path with the police car down to to Rome.

Depending on the parity of the starting position, one of the police cars will catch the car. If the car tries to escape it and pass Rome before the police car reaches it, it won't help because it will then run into the other police car arriving in the other direction. So, whatever the car does, it will bump into a police car and be directed to Rome.