# Answer to Riddle #81: Tracks From A Two Wheeled Bike

81. The diagram shows the tracks from a two wheeled bicycle. Was the bicycle traveling from left to right or right to left?

An example of this appears in The Adventure of the Priory School, a Sherlock Holmes novel. Holmes gets it right of course. But not for the right reasons:

## Identifying the Wheels

So the red line is the front wheel, the blue the back. You may be able to intuit that. The back wheel necessarily follows the front wheel but it is not able to make as tight turns. And the blue line looks like a smoothed version of the red line. But how can we be sure...

The back wheel necessarily points at the front wheel, partly because of the way bikes are made, partly because it just does. At any point in time the direction of a wheel is a tangent to the path it prescribes. The tangent at point 'A' will never intersect the blue line. And so must be the front wheel.

## Determining the Direction of Travel

The bike is traveling right to left...

The key to working out the direction of travel is to realise not only that the back wheel must always point at the front wheel, but it must always be a fixed distance in front of it, i.e. the length of the bike.

In the above diagram the lines BC and EF are the same length, where as CD and FG are not. In fact at any point along the blue line if you take a tangent you will find the red line one bike length ahead of it.

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