I happen to know that Cambridge is a flat and pretty dull place on Sundays (and every other day for that matter). In fact, two friends of mine - John and Jo, the giggling burglars - once went for a drive to brighten up their afternoon. They took their own cars, and drove between the nearby cities of Ayham to Beaton. John took the North Road, and Jo took the South Road. Well... they were more like tracks really - just wide enough for one car (Ayham and Beaton are very small cities).

Checking the map later that day, they discovered that throughout the whole journey, despite all the twists and turns along the roads, they had always been within a mile of each other.

This set them wondering. Was it pure chance, or was there something about the roads which meant that they would always end up within a mile of each other, no matter how carefully they planned their journeys?

They soon realised that if they both started their journey in the same city, they'd already be within a mile of each other at the start of the journey. So they decided that John's journey along the North Road should start at Ayham and travel to Beaton, but Jo's South Road journey should be in the opposite direction, from Beaton to Ayham.

But even with Jo swapping the direction she travels, is it actually possible for John and Jo to complete their journeys without them ever coming within a mile of one another?